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Solutions in Differential Equations

  1. We use linear algebra to solve the linear DE with constant coefficients. See [Pon, pp.95-96, Theorem 10; p.98, Theorem 11] for details. [Arn, p.106, Theorem 15.2] is the final form of the solution. This form suggests some methods for proving the existence theorems in the general case (see [Arn, p.110, Theorem; p.215, Example 2] & [Bir, p.173, Theorem 1; p.175, Corollary]).

  2. Effectiveness.
    1. Closed form solutions.
      1. Inhomogeneous DE: [Bir, p.68, Theorem 5].

      2. By quadratures: Separation of variables & exact equations (integrating factors). The geometrical meaning of exact equations [Sne, p.25].
    2. Approximate solution with (i). error [Bed, p.305, (3)]. (ii). error & derivation [Bir, p.173, Theorem 2].
    3. Stability in terms of energy [Arn1, p.149, Problem 1]. Level sets of the first integral [Arn1, p.140, §12.2 & §12.3] and Liapunov function [Bir, p.128, l.17] are all derivatives of energy. g-u [Pon, p.235, Fig.57(b)] is the projection of ΔE [Arn1, p.150, l.-6] onto the phase plane.

  3. Geometric interpretation of a DE.
    1. y' = v(x) [Arn, p.12, l.7-l.9; Arn1, p.17, l.!6].

    2. F(x,y,y') = 0. p-discriminant equation: Cusp locus, envelope (singular integral), tac locus [Inc, §24-§27].

  4. The method in [Inc, §27] is more effective than that in [Inc, §24-§26].
    1. [Inc, (26.1)] reduces to [Inc, p.75, l.13]. The reduction justifies the nomenclature of "p-discriminant equation".
    2. [Inc, p.75, l.-2; p.77, l.19] show the key difference between a cusp locus and a tac locus.
    3. [Inc, p.77, l.12] confirms the statement in [Inc, p.74, l.7].
    4. The choice of t in [Inc, p.78, l.-3] is more pertinent than that in [Inc, §25].
    5. [Inc, (26.1)] is a discriminant in an algebraic sense [Inc, (27.2)] and is a boundary in a geometric sense [Inc, p.75 (c) & (d)].

  5. If a method is applied to a weaker condition, the application becomes less effective (compare [Inc, §21] with [Inc, '20]). In other other words, to obtain a closed-form solution, applying the method to a DE with a weaker hypothesis will require more drastic cancellations.

  6. In order to find a short cut to solve a DE in practice, [Inc, p.94, l.10-l.21] uses an idea similar to [Bir, p.156, Corollary 1] when a DE has enough integrating factors to recover all its distinct first integrals.

  7. Symbolic operations (Mnemonic devices).
    Symbolic operations enable us to predict and apply a result immediately without thinking of its rigorous proof.
    1. Separation of variables. We use infinitesimals to visualize the symbolic operation and use differential forms to justify its usage [Arn1, p.20, l.5-l.27; p.42, l.-13-p.43, l.8].
    2. Operating Delta-function as a function [Bir, p.55, l.9-p.56, l.3]. Justification: Distribution theory.
    3. Representing an inverse operator as a reciprocal [Inc, p.125, l.-3]. Justification: [Inc, p.126, l.2-l.11].

  8. If we begin by studying the analytic case, we may strike right to the heart of the matter and eliminate complications. Very often we may find a more effective method in the analytic case.
    1. (Rectification) [Arn1, p.89, Theorem 1]. In the analytic case, the Taylor series of a rectifying diffeomorphism can be determined recursively [Arn1, p.91, l.-11].
    2. Differentiable dependence of initial conditions: Trivial.
    3. (Perturbation). It is simple to derive the variational equation in the analytic case [Bir, p.163, l.2-l.15]. In practice, an even simpler method (i.e., Newton's series method [Arn1, p.94, l.17-l.20]) is available.

  9. Effectiveness
    1. Basis for a set of solutions.

    2. Fundamental domain for an individual solution. In order to find the solution of a PDE in a region R, we first find the solution in a fundamental domain I Í R. Then the solution in R\I can be derived from the solution in I by simple algebraic operations [Joh, p.44, Fig.2.5].

  10. Solution's development  
    1. Characteristic Manifolds.
      The characteristic curves of a second-order PDE with 2 independent variables [Joh, p.35, (1.12)] ® The characteristic curves of a first-order system with 2 independent variables [Joh, p.47, (5.13] (See [Joh, p.46, (5.1), (5.2), (5.3)]) ® The characteristic surfaces of a linear first-order system with n independent variables [Joh, p.59, (2.26)] ® The characteristic surfaces of a mth- order quasi-linear system with n independent variables [Joh, p.59, (2.24)].
      Comments.
      1. A solution has its primitive model. To obtain the key idea of solving a generalized problem, we have to trace back to the original setting.

      2. After the key idea is generalized to a certain extent, its spirit and freedom will be developed fully.
    2. Lagrange identities.
          [Sne, p.181, (5)] ® [Sne, p.193, (2)] ® [Joh, p.80, (4.8)].
      Comments. We concentrate on the development of the solution's idea rather than the development of its logic. The first step of the idea development reveals the direction of development, while the first step of the logic development is not very different from the rest of the steps.

  11. The solution pieces together all the information obtained from various approaches.
    1. Point-valued function solution: [Sne, p.145, l.-11]. The difficulty at the singular point [Sne, p.145, l.!6] still remains.
    2. Informal distribution solution: (Potential [Sne, p.145, l.!2]).
    3. Rigorous distribution solution: [Joh, p.92, l.4; p.97, l.-19-l.-15].
    4. (A) is a partial solution of (B): [Joh, p.90, (6.11)].

  12. Existence and uniqueness of the solution of the Dirichlet problem.
    1. A physical and intuitive solution [Sne, p.151, l.19-l.21].
    2. (Not finished).

  13. How the abuse of the supremum property [Bar, p.39, Property 6.4] affects numerical simulation.
        The existence of the function wf [Joh, p.113, (4.9)] comes from the supremum property. It can not provide any significant numerical approximation. Therefore, we must correct our attitude toward axioms: Axioms are designed for understanding a theory's logic network, not for applying the theory. Indiscriminate use (Example: Perron's method [Joh, pp.111-116]) of supremum property will result in assuming that something unknown is known [Wei, p.v, l.-26-l.-25]. This makes the theory less effective. Confucius said, "Do not pretend to know something that you really don't." If our goal is numerical simulation, we must restrict the use of the supremum property to cases where the nonempty set is concretely known and the supremum is numerically approachable.

  14. Solving a PDE involves more than just checking if the given answer is the right solution. Actually, it is the reverse procedure that is rewarding: Suppose the solution exists; what form should it take? In other words, we must use the solution's properties to eliminate the impossibilities and determine its exact form.

    Example 1. For Dirichlet's problem, [Ru2, p.258, l.10-l.13] makes such a mistake: it only checks if the given answer is the solution. We must know the origin of Poisson's integral formula. That is, the Poisson kernel should be recognized as the normal derivative of the Green function [Joh, p.106, l.-25-p.107, l.-8].

    Example 2. The 1-dim wave equation: Solution formula [Sne, p.89, l.-22-l.-13]; Reverse [Joh, p.40, l.16-l.27].

    Example 2'. The 3-dim wave equation: Solution formula [Pet, p.80, l.-13-p.82, l.5]; Reverse [Joh, p.128, l.3-p.129, l.10].

    Example 3. The diffusion equation [Sne, p.283, (6)]: Solution formula [Sne, p.283, (5)]; Reverse [Joh, p.208, l.8-p.209, l.4].

  15. The connection among the various concepts of a solution.
    1. Lagrange's identity [Joh, p.97, (1.17); p.99, (1.23)] simultaneously gives the fundamental solution of the Laplace operator [Joh, p.97, (1.18)] and the solution of Dirichlet's problem [Joh, p.107, (3.7)].
    2. (Solidification by Lagrange's identity [Joh, p.122, l.8; p.123, l.-6]) If a PDE has a point-valued solution, it takes quite an effort [Joh, p.122, l.4-p.124, l.-1] to recognize these distribution solutions as functions [Joh, p.123, (5.26b)]. Because the distribution solution appears in the early stage of solidification, it is quite shallow and vacuous. However, the distribution method gives room for solving PDE's which have no point-valued solutions.
    3. By comparing [Joh, p.97, (1.18)] with [Joh, p.100, (1.31a)], we see that Poisson=s formula [Joh, p.99, (1.28)] is the precursor of the fundamental solution.


  16. An effective method of proving insolvability.

    1. 1-1 correspondence B Reduce to simpler objects.
      Between subfields and subgroups [Her, p.247, Theorem 5.6.6].
      Between Lie subgroups and Lie subalgebras [Po3, p.385, Theorem 83 & p.418, Theorem 91].
    2. A polynomial equation is solvable Þ Its Galois group is solvable [Her, p.255, Theorem 5.7.2].

  17. The spirit of the calculus of variation is to effectively construct a solution using the completeness relations [Cou, p.97, l.-8; p.178, l.12].
        [Cou, pp.97-98, §10.1] and [Cou, pp.174-175, §2.1] propose two solutions to the isoperimetric problem. The former gives an effective method to construct the solution, while the latter does not. [Cou, p.174, l.-12] claims that the set of areas of admissible curves has a least upper bound M [Ru1, p.11, Theorem 1.36]. However, M in [Cou, p.174, l.-11] cannot be practically implemented. This is because E ¹f [Ru1, p.11, Theorem 1.36] implies that the existence of elements in E belongs to the type of logical existence [Wan3, pp.109-110]. We neither know what the elements are nor have enough information to effectively construct the lub using a specified procedure. Thus, in general, the existence of lub in [Ru1, p.11, l.13] is not constructive [Wan3, p.109]. However, for a given specific sequence {a n}, we often have an effective method to construct the lub.

  18. A solution's development (e.g. Green's function associated with a boundary value problem).
    1. Create a solution from physical consideration [Bir, p.53, l.3-l.21].
    2. Generalize the problem by good observations [Bir, p.53, l.-14-l.-7].
    3. Modify the solution in step A accordingly [Bir, p.53, l.-7-p.55, l.8].
    Another example: the forced oscillation [Bir, p.48, Example 6].
    Remark: For the proof of [Bir, p.48, (30)], see [Sym, p.61, (2.207)-(2.210)].

  19. If the method that produces the first solution fails to provide another independent solution, then how do we construct a second independent solution from the first solution?
    1. [Bir, p.36, (13)] ® [Har, pp.50-51, Lemma 3.1].
      Remark. [Har, pp.50-51, Lemma 3.1] gives a general method for reducing a linear system to smaller systems.
    2. (Singularity — using the method of analytic continuation) The case when the roots of the equation [Guo, p.53, (8)] are equal: [Guo, p.55, (17)].
      Special case (regular singularity). The case when the roots of the indicial equation differ by an integer: [Guo, p.61, (27)].
      Example. [Guo, p.224, l.-3].
    3. The Frobenius Method (regular singularity) [Jef, p.483, l.7-l.20]. The case when the two roots of the indicial equation differ by an integer:
      1. The case when the two roots are equal: [Guo, p.62, (8)].
      2. The case when the two roots are not equal (using the properties of a double root) : [Guo, p.63, (13)].
    4. Exploiting the property that the limit of a linear combination of solutions has an indeterminate form.
      Example 1. When u is not an integer, Ju and J-u are linearly independent [Guo, p.348, l.10]. However, when u is an integer n, Ju and J-u are linearly dependent [Guo, p.348, (8)]. In the latter case, we define Yu as in [Guo, p.366, (3)] and then let u®n (the limit of Yu has an indeterminate form).
      Example 2. When u is not an integer, Iu and I-u are linearly independent [Guo, p.374, l.-11]. However, when u is an integer n, Iu and I-u are linearly dependent [Guo, p.374, (5)]. In the latter case, we define Ku as in [Guo, p.374, (6)] and then let u®n (the limit of Ku has an indeterminate form).
      Example 3. [Guo, p.261, l.8].
    5. Changing the contour of the first solution's integral representation.
      Example. Compare [Guo, p.248, (6); p.251, (1)] with [Guo, p.247, (4)].
    6. By inversion [Chou, p.156, (3.148)].
    7. Using the Wronskian [Bir, p.36, l.-9-p.37, l.7].
    8. Neumann's method: [Wat, p.67, l.8-l.10].

  20. [Guo, §2.1-§2.11] discusses how to express the solutions in series form; [Guo, §2.12-§2.14] discusses how to express the solutions in integral form. According to the way that [Guo] is written, these two topics seem to be completely independent of each other. The later topic is mainly based on the method of synthesis. In fact, the domain of a series solution is usually small. We seek an integral solution in order to extend the solution's domain. In the case of Bessel functions, [Wat, p.47, l.-14-l.-13] provides an effective link between a series solution and an integral solution.
    Remark. For a specific-to-general approach (e.g., the series-to-integral approach in [Wat, §3.33]), the reason why each step leads to the next one can be easily explained. The approach is inspirational. In contrast, for a general-to-specific approach, which Landau adopts to make his theory of physics compact, it would very hard to explain why we make a specific choice [Guo, p.352, l.11].

  21. Solving problems through pattern recognition.
        If we want to prove [Mari, p.505, (12.160)] and looking at this problem alone, we may not know where to start. However, if we are familiar with spherical harmonics, we may recognize [Mari, p.505, (12.160)] and [Chou, p.606, (IV.32)] have the same form. Furthermore, [Jack, p.68, (2.35)] is a natural generalization of [Chou, p.606, (IV.32)] (see [Chou, p.606, l.-3]). Therefore, we may prove [Mari, p.505, (12.160)] based on the proof pattern of [Jack, p.68, (2.35)]. Once we recognize the pattern, the proof of [Mari, p.505, (12.160)] becomes trivial.

  22. In [Wangs, p.189, Example], Wangsness leaves his solution in a series form. With this form, we can only evaluate the value of f. In [Jack, p.75, (2.65)], Jackson expresses his solution more neatly in a closed form. With this form, we can know more about the shape and the global properties of the function F.

  23. Green's functions [Bir, p.47, l.11; Ru3, p.192, l.-7-p.193, l.3].
    1. Green's function for the Poisson equation [Jack, p.35, (1.31)].
    2. Green's functions for the wave equation [Jack, §6.4 or [1]].
      Key idea: We use the Fourier transform to remove the explicit time dependence and then use "cause and effect" to eliminate the nonphysical solution.
      Remark. (The wave-particle property) In 1924, Louis de Broglie proposed his postulate for matter waves [Eis, p.56, (3-2)] based on the assumption that matter behaves like radiation. In 1927, G. P. Thomson experimentally discovered electron diffraction and thus confirmed the postulate. Many physicists believed that the research on the wave-particle property is complete once the theoretical assumption is experimentally confirmed. In my opinion, experimental confirmations is far from complete establishment partly because an experiment only represents one particular case and partly because a phenomenon can have many theoretical explanations; one must find the theoretical basis of the wave-particle property in the right place: the solution of the Schrödinger equation. For free particles, the Schrödinger equation reduces to the wave equation [Reif, p.353, l.-3-p.354, l.9]. Only after its theoretical basis is built in the solution of the wave equation from multiple perspectives may we say the wave-particle property is firmly established. The Fourier transforms in [Jack, p.243, (6.33) & (6.34)] are exactly what we are looking for. This is because one can easily prove the  following theorem on the wave-particle property by using the argument in [Coh, Complement EII]:
      [Q, P] = iħ (from the viewpoint of commutators or the Poisson brackets)
      Û [Coh, p.190, (23)] (from the viewpoint of gradients)
      Û the wave function y in ‌p> representation is the Fourier transform of y in ‌q> representation [Coh, p.191, (27)] (from the viewpoint of Fourier transforms).
      One can visualize the wave-particle duality more intuitively using Fourier transforms than using the Poisson brackets.

  24. Effective methods of finding the strong solutions for a wide scope of differential equations
        Modern Mathematicians tend to use the concept of distributions to solve differential equations [Ru3, chap 8]. Sometimes, the solutions will become too weak to be useful for practical problems. If we compare the strength of a generalized solution to the precision of a long-range missile, our goal for solving differential equations should be to aim far and hit the target strongly, precisely and effectively. That is, we are looking for the most effective way to find the strong solutions for a wide scope of differential equations.
    1. Effective methods of integration [Inc1, chap. II]
      1. Exact equations of the first order
        1. Separation of variables [Inc1, §2.11]
          Remark. Homogeneous equations can be solved by the method of separation of variables.
        2. Linear equations of the first order [Inc1, §2.13]
              The homogeneous case can be solved by the method of separation of variables. Using the form of solution in the homogenous case [Inc1, p.20, l.-9], we may solve the nonhomogeneous case by the method of variation of parameters [Inc1, p.20, l.-6].
          Remark 1. The Bernoulli equation can be reduced to the linear form [Inc1, p.22, l.23].
          Remark 2. The Jacobi equation can be reduced to a Bernoulli equation [Inc1, p.23, l.-26].
          Remark 3. The methods of solving differential equations in [Inc1, §2.42, §2.43, and §2.44] are similar to the methods (i) and (ii). Consequently, many textbooks on ordinary differential equations omit these topics even though they have important applications in theoretical physics [Lan1 or Go2].
        3. (Reducing degrees) The Riccati equation can be reduced to a linear equation of the second order [Inc1, p.24, l.-1].
      2. The Clairaut equation [Inc1, §2.45; Sne, chap. 2, §11, (d)]
            The geometric meaning of singular solutions is given by [Inc1, p.40, l.14-l.16; Sne, p.60, l.14].
      3. Reducing the order by integration [Inc1, §2.6 & §2.61]
        Remark. The methods of solving differential equations given in [Inc1, §2.62 & §2.63] are similar to those in [Inc1, §2.6 & §2.61].
      Updates
      1. For Mercator's projection [Inc1, p.34, l.20], see [Kre, p.191, l.4-l.-1].
      2. For [Inc1, §2.6], see [Wid, p.43, Theorem 1].
    2. Linear differential equations of order n
      1. Homogeneous equations with constant coefficients [Edw, §3.3; Pon, pp.50-51, Theorem 5; Cod, p.89, Theorem 6.5]
        Remark 1. The formulation for the properties of the Wronskian determinant given in [Edw, p.162, Theorem 3] is organized, systematic, complete, and useful.
        Remark 2. For the linear independence of [Cod, p.89, (6.20)], the proof given in [Pon, p.54, l.-24-l.-8] is clearer than that given in [Cod, p.90, l.4-l.20].
        Application.
            Coddington's observation regarding similarity [Cod61, p.149, l.9] is inadequate. Actually, Euler's equation can be transformed into an equation with constant coefficients [Col, p.110, l.-19-p.120, l.14]. While [Col, chap. II, §6] links Euler's equation to its indicial equation [Har, p.85, l.14], [Har, p.85, l.5-l.11] further links Euler's equation to the Jordan normal form of the corresponding linear system [Har, p.84, (12.3)]. The statement given in [Har, p.85, l.7-l.8] can be proved by [Har, p.59, (5.18)]. 
      2. Nonhomogeneous linear differential equations
        1. The method of undetermined coefficients [Edw, p.191, l.-3-p.201, l.9].
           Remark. Given L[u] = f(x), where L[u] is a linear differential operator. Let B={ba} be a basis. If ("baÎB,  there exist finite i such that L[ba] = åi cibi), we may find a solution by substituting u(x) = åa daba into L[u] = f(x). (Proof. Express f(x) as åa haba.) We shall use the rules of the method of undetermined coefficients to check if B is an appropriate basis. Examples. We may let B be {xn | n ³ 0}, {cos nx, sin nx} or {Jn, Yn}. Similarly, for a regular Sturm-Liouville system (L=D[p(x)D]-q(x)) [Bir, p.256, l.-8], we may find a solution u(x) = åa daba (where {ba} are eigenfunctions of L[u]) of L[u] = f(x).
        2. Variation of parameters: [Edw, p.201, l.10-p.203, l.-1] ® [Cod, p.87, Theorem 6.4].
    3. Linear systems
      1. Homogeneous systems with constant coefficients [Har, chap. IV, §5]
        Remark. We use [Jaco, vol. II, p.94, (27); p.97, (29)] to choose Q so that J is in a Jordan normal form [Har, p.58, l.-4]. Linear algebra will reach a dead end after it finishes discussing Jordan normal forms. Only after we go further into the theory of ordinary differential equations may we use Jordan normal forms to obtain the solutions of differential equations.
         Application.
             The case of variable, but periodic, coefficients can theoretically be reduced to the case of constant coefficients [Cod, p.78, Theorem 5.1; Har, p.60, Theorem 6.1].
      2. Nonhomogeneous systems (variation of constants): [Har, pp.48-49, Corollary 2.1].
      Remark. We discuss first the solutions of linear differential equations of order n, and then the solution of general linear systems. The former solutions are more effective than the latter ones. [Cod, chap. 3] and [Har, chap. IV] reverse the order of presentation. I do not think their approach is appropriate.
    4. (Transformed Bessel Functions) Using the transformation given in [Edw, p.582, (2) and (5)], we may transform the Bessel equation of order p
      z2(d2w/dz2)+z(dw/dz)+(z2-p2)w=0 into the equation
      x2y"+Axy'+(B+Cxq)y=0 [Edw, p.582, (3)]. Moreover,
      using the transformation [w = x(a-1)/2 exp[(b/p)xp]y, z = xq, a = (1-a)/2, g = (2-r+s)/2,
      l = 2(|a|)1/2/(2-r+s), and n = [(1-r)2-4b]1/2/(2-r+s)], we may transform the Bessel equation
      z2(d2w/dz2)+z(dw/dz)+(l2z2-n2)w=0 into the equation
      x2y"+x(a+2bxp)y'+[c+dx2q+b(a+p-1)xp+b2x2p]y=0 [Barr, p.586, l.3].
      Remark. We must prove [Barr, p.586, Theorem 1] using the proof of [Edw, p.583, Theorem 1] as our model. In other words, we must be familiar with the technique of solving a simple case before we attempt to solve a more difficult problem.
    5. Using the continued fraction method to solve the DE's with three-term recurrence relations
      Examples: the tidal equation [Jef, §16.08], Mathieu's equation [Jef, §16.09], and equations with infinite determinants [Jef, p.489, l.3].
    6. Using the adjoint operator to represent the solution in the complex integral form [Guo, §2.12]
      1. The Laplace transform [Jef, §16.10; Guo, §2.13]. The main feature of the Laplace transform method is the integration by parts [Jef, p.489, l.-4]. The Laplace transform method is useful for finding the asymptotic expansion of the solution [Guo, p.84, l.-8].
        Examples: Bessel's equation [Jef, p.490, (15) & (16)] and the Hermite equation [Guo, p.84, (17)].
      2. The Euler transform [Guo, §2.14].
      3. The Mellin transform [Guo, p.80, (18)].
    7. Sometimes, the differential equation of the inverse function t(s) is simpler than the equation of motion for s(t) [Cou2, vol. 1, §4.5. §4.7.a, §4.7.b].

  25. Green function expansion in cylindrical coordinates [Chou, §3.14; Jack, §3.11]
        In view of [Chou, p.166, (3.176) & (3.177)], if we replace d(f-f') and d(z-z') in [Chou, p.166, (3.175)] by Q(f)Q*(f') and Zn(z)Zn*(z') respectively, then [Chou, (3.175)] will reduce to [Chou, p.167, (3.181)]. Thus, by fixing m and n and using separation of variables we may reduce the 3-dimensional boundary value problem to a one-dimensional boundary value problem of the Sturm-Liouville type. Then it is obvious that
    G(r, r')= Sm Sn gmn(r|r')Q(f)Q*(f')Zn(z)Zn*(z') will be the solution of [Chou, p.166, (3.175)]. Both Jackson's proof of [Jack, (3.141)] and Choudhury's proof of [(3.179)-(3.182)] are unnecessarily complicated.

  26. The equivalence of a differential equation and its corresponding integral equation [Coh, p.915, l.-11-p.916, l.9].

  27. Solutions of partial differential equations
    Complete solutions, general solutions, and singular solutions ([Sne, p.49, l.-21-l.-5] ® [Sne, p.59, l.-15-p.60, l.15])
        We try to find the smallest geometric entity which satisfies [Joh, p.9, (4.2)]. A curve is inadequate as a solution because a line, a typical curve, has d/dt but cannot have both /x and /y. Because a surface has a higher dimension, we try to use it [Joh, p.9, l.15] as the building block of our solutions. In view of [Inc1, p.4, l.-16-p.5, l.17], it is natural to consider the 2-parameter family of surfaces given in [Sne, p.60, l.4] as complete solutions.
    Remark. [Sne, p.59, l.-15-p.60, l.15] should be supplemented by [Wea1, vol. 1, §15 & §21] in order to gain a complete understanding.
    1. Examples [Cou, vol. II, chap. 1, §1.1].
          There are two reasons that Courant designed various examples: First, mastering various techniques of solving PDEs will enhance one's ability to solve any PDE. Second, theorems in the theory of PDEs come from induction based on one's observations when studying examples. Without examples, it is difficult to find a theorem's origins and key points.
    2. dx/P = dy/Q = dz/R [Sne, chap. 1, §3].
    3. Pdx+Qdy+Rdz = 0
      1. Solutions of integrable equations
        1. Geometric meanings: [Sne, p.25, l.6-l.26; l.27-l.30].
        2. A necessary and sufficient condition for integrability [Sne, p.21, Theorem 5; Inc1, §2.8].
          Remark. The fact that the condition of integrability is a sufficient condition provides a general method of solving a Pfaffian differential equation [Sne, p.19, (6)].
        3. Integrability vs. accessibility: Carathéodory's formulation [Sne, p.33, l.-20-p.37, l.14]; Born's formulation [Sne, p.37, l.-20-p.38, l.-1].
          Remark 1. For a non-integrable equation, we must project the path onto a 2-dim vector space in order to make the concept of accessibility meaningful [Sne, p.36, Fig. 11; p.38, Fig. 12].
          Remark 2. Carathéodory's proof stops at the equation given in [Sne, p.37, l.13], a condition which is equivalent to integrability [Sne, p.35, l.-10-l.-5]. Born's proof proceeds further by directly constructing the integral surface.
        4. Effective methods of solving integrable equations [Sne, chap.1, §6].
          Remark. Among the methods given in [Sne, chap. 1, §6], both Method (e) and Method (f) apply to the general case, while the remaining methods apply only to special cases.
        5. Carathéodory's  version of the laws of thermodynamics: Carathéodory puts the laws of thermodynamics on a firm mathematical foundation.
          (1). The first law [Sne, p.39, l.17-l.21]: The first law defines internal energy [Sne, p.40, (1)] based on [Ru1, p.115, Theorem 6.16] and also defines the concept of heat [Sne, p.40, (2)]. It shows that for two variables we do not need the second law [Sne, p.40, l.-6-p.41, l.6] to find an integrating factor m(p, v) for DQ [Sne, p.40, (4)]. Thus, in term of mathematics, integrability is the crucial line dividing the first law and the second law.
          (2). The second law [Sne, p.41, l.26-l.27]: The concept of reversible path given in [Hua, p.17, Fig. 1.7] is not effective because [Hua] fails to provide an effective test to check whether the heat in the Pfaffian differential form is integrable. Carathéodory's  version of the second law [Sne, p.41, l.26-l.27] does provide the most effective test based on [Sne, p.19, Theorem 2]. In order to define entropy, the approach using reversible paths to seek an integrating factor, as Huang employed in [Hua, §1.4], is good for two independent variables, but is inadequate for more than two independent variables. The use of accessibility [Sne, p.34, l.15-l.29] is the only way to make the concept of entropy well-defined [Sne, p.35, Theorem 8]. The path used to build accessibility runs both ways, so it is always reversible. Furthermore, there are too many physical versions of the second law [Reic, §2.D.3]. Carathéodory's  version reveals the mathematical insight which implies all these phenomena. The fact that heat cannot flow from a cold body to a hotter one is an example of inaccessibility because the two states are not on the same integral surface. [Zem, §7-7 & §7-8] fail to prove important theorems [Sne, p.19, Theorem 2; p.35, Theorem 8] and emphasize only trivial gimmicks. Consequently, after reading these two sections, readers will remain puzzled about Carathéodory's  version of the second law.
          Remark. Textbooks should not be used to show off one's pedantry by giving new terminology without any references. They should be written with an audience in mind and should make it easy for readers to understand the essence of the topic.
      2. Solutions of non-integrable equations: [Inc1, §2.83; Sne, p.26, Fig. 10].
    4. The first order
      1. Linear
        1. Analytic solutions: [Sne, p.50, Theorem 2; p.53, Theorem 3].
              The geometric meaning of general solutions is given by [Inc1, §2.71; Sne, p.50 Theorem 2]. An integral surface [Joh, p.9, l.16] represents a general solution of the partial differential equation [Sne, p.50, (1)]. The general solutions of the corresponding system of ordinary differential equations [Sne, p.50, (4)] are 2-parameter families of characteristic curves [Sne, p.51, (5)]. These two solutions are identical [Joh, p.10, l.7-l.15; Inc1, p.50, l.15-l.28]. [Inc1, .50, l.-9-p.51, l.-9] proves [Sne, p.50, (3)] which allows us to form an integral surface [Inc1, p.48, l.-18-p.49, l.15] by selecting a one-fold infinity of curves of congruence from the two-parameter family of curves [Sne, p.51, (5)].
          Remark. [Joh, p.10, l.-6-p.11, l.3] discusses the geometric consequence of [Sne, p.50, l.-10-l.-6].
        2. The method of finite difference [Joh, p.7, l.-8-p.8, l.12]
              If we use backward differences, the error will not grow exponentially with the number of steps in the t-direction [Joh, p.7, (3.17)]. The Courant-Friedrichs-Lewy test is satisfied when [Joh, p.7, (3.16)] is satisfied.
      2. Nonlinear
        1. Cauchy's method of characteristics
          (1). The initial value problem is a special case of the Cauchy problem [Joh, p.11, l.-7-p.12, l.2].
          (2). Linear [Ches, p.9, l.9-l.21]® quasi-linear [Joh, p.12, l.3-p.13, l.18]® general [Sne, p.65, l.12-l.-4]. If one jumps directly to the third step without going through the first and the second, it would be difficult to recognize the method's structure, scheme, and key points.
          (3). It is easier to grasp the key idea in a simple case [Ches, p.3, (1-6) or p.6, (1-16)]. We choose two natural parameters for the integral surface passing through a given curve G(s): one from the parameter s of the curve G and the other from the parameter t of the characteristic curve passing the fixed point G(s) [Joh, p.9, (4.4)]. [Joh, p.12, l.3-p.13, l.13] proves the local existence of the solution of the Cauchy problem for the quasi-linear case [Joh, p.9, (4.2)]. The definition of characteristics given in [Joh, p.9, (4.3)] is easily generalized and visualized. In contrast, the definition given in [Ches, p.4, l.14] is too simple to be generalized, while the definition given in [Sne, p.64, (18) or p.69, (6)] is too complicated to be visualized. The reason that five equations are required to determine the characteristics in [Sne, p.69, (6)] is because the denominators in the first terms in [Sne, p.69, (6)] involve not only {x,y,z} but also {p,q} [Ches, p.175, l.-16-l.-6; p.176, l.-2-p.177, l.11].
          (4). The characteristic equations given in [Sne, p.64, (18)] reduce the original PDE [Sne, p.62, (2)] to a normal system of first-order ODEs.
          (5). The language in [Sne, chap. 2, 8] is very loose, in the following I try to make the definitions more precise and match their geometrical meanings more closely:
          (i). A plane element satisfying F(x,y,z,p,q)=0 is an integral element.
          (ii). A one-parameter (t) family of plane elements [Sne, p.63, (7)] satisfying [Sne, p.63, (8)] is a strip containing the curve C(t).
        2. Charpit's method [Sne, chap. 2, §10].
              [g(x,y,z,p,q)=0 and f(x,y,z,p,q)=0 are compatible] Û ((f,g)/(p,q) ¹ 0 and [f,g]=0)
                                                                            [Sne, p.69, l.-9].
          (a). f(p,q)=0 [Sne, p.71, (1)].
          (b). f(z,p,q)=0. [Sne, p.71, (5)].
          (c). f(x,p)=g(y,q) [Sne, p.72, (7)].
          (d). Clairaut equations: z = px+qy+f(p,q) [Sne, p.72, (9)].
          Remark 1. [Sne, p.67, (4)] can be derived from the implicit function theorem [Wid, p.59, Theorem 16].
          Remark 2. [f, g] given in [Sne, p.68, (9)] is a generalization of the Poisson bracket [Ches, p.215, l.-1].
          Remark 3. Cauchy's method and Clairaut's method reach the same conclusion [Sne, p.70, l.1-l.3] through different paths.
        3. Jacobi's method
        Remark 1. Any general first-order system of ODEs can be written in the form of a Hamiltonian system [Ches, §8-19].
        Remark 2. [Ches, p.194, Theorem 3] can be proved using envelopes and characteristic curves [Ches, p.194] or using generating functions F(2) [Ches, p.208, l.-13]. The equation given in [Ches, p.208, (8-177)] is incorrect. It should have been -H(x,t,p) = Ft(2).
      3. Solutions satisfying given conditions [Sne, chap. 2, §12]
        1. An integral surface passing through a given curve C is the envelope of the set of those members of the family [Sne, p.73, (3)] that touch the curve C.
        2. An integral surface circumscribing a given surface S is the envelope of the set of those of members of the family [Sne, p.73, (3)] that touch the surface S.

  28. Properties of solutions of ordinary differential equations (dx/dt = V(t,x), where t represents time and x represents a point in Rn)
    1. Uniqueness theorem [Bir, p.142, Theorem 1].
    2. Continuity theorem [Bir, p.143, Corollary (b)].
    3. The global existence theorem for the case that X satisfies the Lipschitz condition globally [Bir, p.152, Theorem 6].
      Remark. [Arn1, p.269, Fig. 215; p.274, Fig.221 & Fig.222] give a geometrical interpretation of Picard's successive approximation. Note that [Arn1, p.269, Fig. 215] provides more information than [Bir, p.153, Fig. 6.1].
      1. The global existence theorem for linear systems [Bir, p.156, Theorem 7 & Corollary 1].
        Remark. In the proof of [Spi, vol.1, p. 228, Proposition 17], Spivak made one mistake in [Spi, vol.1, p. 228, l.6] and a second in [Spi, vol.1, p. 229, l.5]. Although the proof of [Arn1, §27.2] corrects Spivak's mistakes, Arnold's local approach reduces the solution's effectiveness in [Arn1, p.101, l.21].
        1. Linear equations with constant coefficients [Arn1, p.164, the fundamental theorem].
    4. The local existence theorem for the case that V satisfies the Lipschitz condition locally [Bir, p.157, Theorem 8].
      Remark. The proof given in [Arn1, chap.4, §31.8, p.276, Corollary] is well organized: it utilizes mathematical structures neatly.
    5. The local existence theorem for the case that V is continuous locally [Bir, p.166, Theorem 13].
    6. Smoothness of solutions
      1. If V(t,x) is of class p with respect to t [Kre, p.21, l.11-l.19], then any solution g is of class p+1 with respect to t [Pet66, p.54, Theorem].
      2. If V(t,x) is analytic with respect to t, then any solution g is analytic with respect to t [Bir, p.160, Theorem 9 and its corollaries]. If V(t,x) is analytic on 0<|t|<a, then any solution can be represented as g(t) = Z(t)tR, where Z(t) is single-valued and analytic on 0<|t|<a [Har, p.70, Theorem 10.1].
        Remark. The statement given in [Guo, p.136, l.4-l.3] is derived from b.
    7. Continuity and differentiability of the solutions
      1. Continuous dependence on parameters (local: [Pon, pp.170-171, Theorem 13]; global: [Pon, p.194, Theorem 16]).
      2. Continuous dependence on V (local: [Bir, p.147, Corollary])
      3. Differentiability with respect to the parameters (local: [Pon, p.173, Theorem 14]; global: [Pon, p.197, Theorem 17])
      4. Continuous dependence and differentiability of a solution g with respect to the initial points x (local:
        (i). (VÎC1 in a neighborhood of (t0, x0)) Þ gÎCx1 [Pon, pp.179-180, Theorem 15].
        (ii). (VÎCr in a neighborhood of (t0, x0)) Þ (gÎCr in t and x jointly), where r = 1, 2, … and ¥ [Arn1,p.285, l.12];
        global: [Pon, p.198, Theorem 18]).
        Remark 1. (ii) is a corollary of (i). See [Arn1, p.285, l.15-l.16] or [Pon, p.177, l.-20-l.-1]. For the proof of (i), read [Pon, p.179, l.-13-p.181, l.11] instead of the confusing proof given in [Arn1, p.285, l.16-p.287, l.-1] [1].
        Remark 2. [Pon, p.173, (15)] motivates us to define the system of equations of variations given in [Arn1, p.279, (2)].
        Remark 3. [Arn1, p.279, Fig.226] gives a geometrical interpretation of the system of equations of variations.
    8. Solutions represented by phase flows [Arn1, chap.1, §4]
           [Spi, vol. 1, p.203, Theorem 5] motivates us to formulate a clean definition of the phase velocity vector of a phase flow given in [Arn1, p.63, Definition]. However, the notations given in [Spi, vol. 1, p.202] are very confusing, so it is difficult to recognize the conclusion given in [Arn1, p.63, Definition]. Furthermore, [Arn1, chap.1, §4] provides concrete examples of phase flows.
    9. The rectification theorem (statement: [Arn1, p.89, Theorem 1]; proof: [Arn1, pp.283-284])
      Remark 1. The proof given in [Arn1, pp.283-284] emphasizes the insight. The proof given in [Spi, vol. 1, pp.205-207] emphasizes calculations. The proof given in [Bir, p.165, Theorem 1] emphasizes the shortcut to the conclusion. Consequently, if one reads the third proof without reading the first one, it would be difficult to understand the proof and catch the essence of the theorem.
      Remark 2. Arnold uses the rectification theorem to create a trivial example of differential equations. Then he uses this trivial example to discuss basic theorems in differential equations. See [Arn1, chap. 2]. In my opinion, for the basic theorems in differential equations, we should discard [Arn1, chap. 2] and retain only [Arn1, chap. 4]. This is because Arnold fails to discriminate among the effectiveness of various existences. In addition, his construction of a rectifying diffeomorphism is based on the assumption that the differential equation is solved. Thus, he fails to discuss all the effective methods of solving differential equations. In addition, each theorem in differential equation arises from the need to solve mathematical or physical problems and has its own typical example to represent its significance. When we have not yet encountered that problem or example, the discussion of the theorem using a trivial example simply diminishes or obscures its meaning.
    10. Adjoint equations: Green's formula and the Lagrange identity [Har, p.66, l.-12-p.67, l.5] [1].
      Remark. The proof given in [Har, p.66, l.-6-l.-3] is simpler than the proofs given in [Cod, p.86, l.8-l.-6].
    11. Stability for plane autonomous systems: [Inc1, §6.6] ® (linear [Bir, chap. 3, §5]) ® (nonlinear [Bir, chap. 5, §7-§9]).
      Remark. In order to see the big picture, we must simplify the setting, stress the main point, use fewer definitions, and ignore minor differences. [Inc1, §6.6] does not formally define stability, but it provides a typical and an intuitive case of stability by detaching the concept of stability from differential equations. [Inc1, §6.6] also shows us how to produce stability. In contrast, [Bir, p.121, Definition] tries to distinguish the nuances among various stabilities, but ends up only obscuring the big picture of the concept.¬
    12. The order of any determinate system of linear equations with constant coefficients is equal to the order of its characteristic equation [Inc1, p.150, l.24-l.26].

  29. The solutions of ordinary differential equations that cannot satisfy the given boundary conditions
    Example. [Sag, p.65, l.8].

  30. Formal solutions vs. actual solutions
    1. Singularities of the first kind: The formal solution converges to an actual solution [Cod, p.117, Theorem 3.1].
      Remark 1. z0 is an "at most singularity of the first kind" if and only if z0 is a regular singularity [Cod, p.124, Theorem 5.1; p.125, Theorem 5.2].
       Remark 2. [Was, p.28, l.10-l.-1] shows that the regular singularity of a single differential equation of nth order is a special case of the regular singularity of a system of linear differential equations.
      Remark 3. Formal solutions must be inclusive. For formal solutions, the formal power series given in [Har, p.78, Theorem 11.3] is inadequate for representing typical solutions. Consequently, we should consider formal logarithmic sums [Cod, p.116, l.13]. If we examine the proofs of [Cod, p.119, Theorem 4.1] and [Cod, p.121, Theorem 4.2], it is unnecessary to use [Cod, p.117, Theorem 3.1] to prove that the above two theorems are valid if we just consider F as formal solutions rather than actual solutions. With this prerequisite knowledge, then for the proof of [Cod, p.117, Theorem 3.1], it will suffice to consider only these reduced formal solutions given in [Cod, p.121, Theorem 4.2].
    2. Singularities of the second kind: There exists an actual solution such that it has the formal solution as its asymptotic expansion [Cod, p.160, Theorem 4.1; chap. 5, sec. 6].
      Remark 1. [Col, p.255, l.3-p.256, l.10] provides a complete proof of the indicial equation.
      Remark 2. [Cod, p.152, l.-15-l.-6] The essence of [Cod, p.160, Theorem 4.1] lies in [Cod, p.155, Lemma 4.2]. Even though the formal solution can be divergent [Cod, p.139, l.11], we may construct an actual solution in the integral form using the variation of constants [Cod, p.155, l.18-l.21]. Then the error between the actual solution and the partial sum of the expansion can be estimated by the successive approximations [Cod, p.156, (4.25)].
      Remark 3. The second line of [Cod, p.156, (4.32)] should have been "R(qi(t)-ql(t)) is bounded above" rather than "R(qi(t)-ql(t)) is nonincreasing". Levinson has made a mistake here.

  31. The domain of the solution of a system of ordinary differential equations is closed and connected [Har, p.15, Theorem 4.1].

  32. Peano's existence theorem [Har, p.10, Theorem 2.1]
    1. The definition of ye given in [Har, p.10, l.-8-l.-1] is clear, while the definition of xn given in [Bir, p.166, l.-6-l.-4] is not. Clarity should be the top priority of a textbook.

    2. When we propose a new theorem that is similar to a theorem that already exist in a theory, we must describe their sameness as completely as possible and then identify their crucial differences. The comparison helps us identify the unique feature of the new theorem. The comparison of the pair [Bir, p.152, Theorem 6; p.166, Theorem 13] is less complete than that of the pair [Har, p.8, Theorem 1.1; p.10, Theorem 2.1].

  33. Regular singular points
    1. Definitions
      1. (Theoretical definitions) For a linear system, the definition of regular singular points is given by [Har, p.73, l.7]. A differential equation of dth order [Har, p.84, (12.1)] has a regular singular point at t=0 if [Har, p.84, (12.3)] has a regular singular point at t=0. The equation and the system are related by [Har, p.84, (12.2)].
      2. (Practical definitions) [Har, p.84, (12.1)] has a regular singular point at t=0
        Û The solutions of [Har, p.84, (12.1)] are linear combinations of tl(log t)ka(t), where a(t) is analytic for |t|<a [1]
        Û [Har, p.84, (12.3)] has a simple singularity [Har, p.84, l.13-l.16] at t=0 [Har, p.85, Theorem 12.1].
      Remark 1. For singularities at infinity, see [Cod, p.128, Theorem 6.1 & Theorem 6.2].
      Remark 2. It would be difficult for one to see the big picture if one studies only the definition given in [Bir, p.233, Definition] or [Guo, p.56, l.-8].
    2. The Frobenius Method
      1. When the roots of the indicial equation do not differ by an integer [Bir, p.234, Theorem 5; p.240, Theorem 7]
      2. When the roots of the indicial equation differ by an integer [Bir, p.242, Theorem 8].
      Remark 1. Coddington's approach [Cod61, p.158, l.-9-l.-7] must deal with the minus signs [Cod61, p.146, l.-6-p.147, l.3] that open Pandora's box. The way out of this mess of entanglement is to treat zr as exp [r (log z)].¬
      Remark 2. [Wat1, §10.32] gives an alternative method to solve a differential equation with a regular singular point. Since the equality in [Wat1, p.201, l.4] links to the difference of the exponents, the solution given in [Wat1, p.201, l.3] is more informative than the solution given in [Bir, p.242, l.-10].
    3. Fuchsian equations [Cod, p.129, l.8]
      The principle of reducing differential equations to a simple form: A general linear fractional transformation transforms one set of regular singular points into another set of regular singular points, and the indicial equations of the transformed DE coincide with those of the original DF at corresponding points [Bir, p.251, l.-18-l.-15].
      1. First-order [Bir, p.251, l.6-l.10].
      2. Second-order Fushian equations with two regular singular points can be reduced to the Euler DE [Bir, p.251, l.-7].
        Remark. The solutions given in [Wat1, p.208, l.-14-l.-11] are derived from [Bir, p.251, l.-8; Col, p.110, l.12; p.111, l.12].
      3. Second-order Fushian equations with three regular singular points can be reduced to the hypergeometric DE [Bir, p.253, Corollary; Guo, §2.9; Cod, p.132, (7.8)]. Example: the associated Legendre equations [Guo, p.71, (15)].
        Remark 1. In their attempt to find the required transformation, [Bir, p.253, l.-15-l.-5] provides the method without giving the final answer, while [Guo, p.69, (6); p.70, (11)] provides the final answer without explanation. It is simpler to derive [Guo, p.70, 12] by means of [Bir, p.251, l.-18-l.-15] rather than from the general case [Guo, p.70, (11)].
        Remark 2. The proof of the Riemann identity given in [Bir, p.253, (41)] is simpler than that given in [Guo, p.68, (2)] because the latter considers the general case, while the former considers the special case, the Riemann DE [Bir, p.252, (40)]. Since the indicial equation of the general case remains the same as the indicial equation of the special case by a linear fractional transformation, it is unnecessary to prove the Riemann identity in the general case [Bir, p.251, l.-18-l.-15].
      4. Second-order Fushian equations with five regular singular points [Guo, §2.8].
      5. Second-order Fushian equations with k regular singular points [Guo, §2.7] [1].
        Remark. The indicial equation at infinity given in [Bir, p.250, (39)] is simpler than that given in [Guo, p.65, (6)] because the former considers only one regular singular point z = ¥, while the latter considers the combination of the regular singular point z = ¥ and other k regular singular points.
      6. Higher order equations [Cod, p.129, Theorem 6.4]
      7. Linear systems [Cod, p.129, Theorem 6.3]

  34. Comparing solutions' Effectiveness
    1. (Picard's successive approximations) [Cod, p.22, l.1-l.5]
    2. (Formal solutions near a simple singularity [Har, p.73, l.20]) [Cod, p.117, Theorem 3.1]
    3. (Power series solutions near an ordinary point) [Cod61, p.129, Theorem 12]
        Suppose we want to solve a differential equation such as [Edw, p.528, (1)]. Then which of above methods shall we use? If our goal is to show the existence of a solution, then method A is adequate. Suppose we want to use an effective method to find the solution. For method A, we must compute integrals, while for method B and method C all we have to do is compare corresponding coefficients. Therefore, for computations the latter two methods are simpler. In addition, for method C we need consider only analytic (ordinary) points, while for method B we have to expand our consideration to simple singularities. Furthermore, method B shows that the solution is in the form of formal logarithmic sums, while method C shows that the solution is in the form of power series. The latter is more specific. In summary, in order to find effective solutions, our computations must be simple; our consideration should be focused on as small a scope as possible; the form of solutions must be specific.¬

  35. In [Wat1, p.478, l.10], the solution of the differential equation given in [Wat1, p.478, l.12] is expressed in terms of the Theta functions. In [Gon1, p.421, (5.17-1)], the solution of the differential equation given in [Gon1, p.425, (5.17-7)] is expressed in terms of Tan. In the above two cases, we use more familiar functions such as Tan and the Theta functions as our tools to make the solution sn tangible and constructive. The attempt to link the solution to the functions with which we are familiar leads to many new questions. For example, how do we find t if k is given? The answer is given in [Wat1, §21.7, §21.71 & §21.711]. In contrast, sn u in [Guo, p.530, (1)] is defined by the differential equation given in [Guo, p.931, l.-7]. The solution is less tangible, but we try to find its meaning [Guo, §10.2] and establish its properties such as the addition formula without depending on other familiar functions [Guo, §10.4].
    Remark. Treating differential equations as a isolated subject is a superficial approach. Solving a differential equation involves solving a group of differential equations. In order to fully understand the solution of a differential equation, we must study it from multiple perspectives.

  36. The introduction of generating functions facilitates the operation of solutions, e.g., establishing recurrence relations [1]. As another example, the use of generating function simplifies the proof of orthonormality relations for a certain measure (compare [Guo, p.324, l.-2-p.325-l.7] with [Wat1, p.350, l.-11-p.351, l.12]). Furthermore, we use Laurent's theorem [Wat1, §5.6] to produce generating functions in most cases [Guo, p.323, (13); Wat1, p.355, l.16].


  37. Laplace's equation
    1. There are at most 2n+1 independent solutions of degree n [Wat1, p.389, l.12-l.17].
    2. gm(X,Y,Z) is an even function of Y, while hm(X,Y,Z) is an odd function of Y.
      Proof. Z + iX cos u + i(-Y) sin u = Z + iX cos u+ iY sin (-u).
    3. gm(0£m£n) and hm(1£m£n) are independent solutions of degree n [Wat1, p.389, l.-11-l.-6]
    Remark 1. We may use the same method prove the following theorem about PDE:
    Let {fn(u)} be an orthonormal basis and V0(x,u) = Sn=1¥ gn(x) fn(u) be a particular solution of P(D)V(x)=0.
    Then P(D) gn(x) = 0 (n = 1,2,3,…).
    Remark 2. By mathematical induction, cos h q sin k q = S m=0h+k (Am cos mq + Bm sin mq).


  38. Solving linear equations of the second order using continued fractions [Inc1, §7.5]
    Remark. The statement given in [Inc1, p.179, l.-17-l.-10] can be proved by [Perr, p.292, Satz 46D; p.24, Satz 2].
    39. The proof of [Perr, p.292, Satz 46D] uses [Perr, p.285, Satz 41]. The proof of [Perr, p.285, Satz 41] uses [Perr, pp.285-286, Satz 42; p.276, Satz 38]. The proof of [Perr, pp.285-286, Satz 42] uses [Perr, p.262, Satz 30] and [Perr, pp.280-281, Satz 40].
    Remark. [Inc1, §7.501] provides a constructive method of solving the ODE given in [Inc1, p.179, l.-7].


  39. Links {1}.