## Abstract

*A differential equation* is an equation involving unknown functions, independent variables and derivatives (or differentials) of unknown functions.

## Keywords

General Solution Singular Point Characteristic Equation Arbitrary Constant Homogeneous Equation
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## Notes

- (1)See p. 394.Google Scholar
- (1).Complex values of functions are also taken into account.Google Scholar
- (1).More precisely, the assumptions of Cauchy’s theorem are not fulfilled at any point such that
*cx*+*ey*= 0; however, in this case we may interchange the variables and consider the equation \(\frac{dx}{dy}=\frac{cx+ey}{ax+by}\) which satisfies these assumptions.Google Scholar - (2).
- (1).
- (1).This method can also be applied in the case, when
*F*(*x*) =*Q*_{p}(*x*), i.e., when*k*= 0, and also in the cases, when*F*(*x*) =*Q*_{p}(*x*)*e*^{rx}cos ω*x*or*F*(*x*) =*Q*_{p}(*x*)*e*^{rx}sin ω*x*which corresponds to the value*k*=*r*±*i*ω. In the latter case we should seek for a solution in the form \(y=x^{m}e^{rx}\;[M_{p}(x)\;\textup{cos}\;\omega x+N_{p}(x)\;\textup{sin}\;\omega x]\).Google Scholar - (1).
- (1).
- (2).\(\frac{d^{n}\varphi} {dt^{n}}\) is assumed to satisfy the conditions introduced above under which the transform is defined.Google Scholar
- (1).The case when the determinant is zero requires additional consideration.Google Scholar
- (1).For the Γ function, see p. 192.Google Scholar
- (2).
- (1).The convergence of the hypergeometric series (1), for
*x*= 1 or*x*= −1, depends on the number δ = γ − α − β. For*x*= 1, series (1) is absolutely convergent, if δ>0, and divergent, if δ ⩽ 0; for*x*= −1, the series is absolutely convergent, if δ > 0, conditionally convergent, if − 1<δ⩽0, and divergent, if δ⩽ − 1.Google Scholar - (1).We shall assume that the interval (
*a*,*b*) is finite. In the case of an infinite interval, the results change essentially.Google Scholar - (1).In solving a system in this form, any of the variables
*x*_{k}for which*X*_{k}≠0 can be taken as the independent variable. Then the system assumes the form \(\frac{dx_{j}}{dx_{k}}=\frac{X_{j}}{X_{k}},j=1,2,\cdots, n\). It is more convenient, however, to preserve the symmetry and introduce the parameter*t*as a new variable, letting \(\frac{dx_{j}}{X_{j}}=dt\) or \(\frac{dx_{j}}{dt}=X_{j}\).Google Scholar - (1).If at least two positive and at least two negative coefficients occur in the equation, then the equation is called
*ultrahyperbolic*.Google Scholar - (1).In this formula, the symbol
*x*is used to denote the system of*n*variables*x*_{1},*x*_{2}, …,*x*_{n}regarded as a point of the*n*-dimensional space and*L*[*u*] is a linear differential expression which may involve the derivative \(\frac{\partial u}{\partial t}\) but which does not involve derivatives of higher order with respect to*t*.Google Scholar - (1).Δ is the Laplace’s operator with respect to
*n*variables*x*_{1},*x*_{2}, …,*x*_{n}(see p. 566).Google Scholar - (2).
- (1).Δ is the Laplace operator with respect to
*n*variables*x*_{1},*x*_{2},…,*x*_{n}(see p. 566).Google Scholar - (1).It follows from the following discussion that the boundary conditions cannot be satisfied, when this constant is positive.Google Scholar
- (1).The equation
*conjugate*to the linear equation \(\sum_{i,k}a_{ik}\frac{\partial ^{2}u}{dx_{i}dx_{k}}+\sum_{i}b_{i}\frac{\partial u}{\partial x_{i}}+cu=f\) is the equation \(\sum_{i,k}\frac{\partial ^{2}(a_{ik}v)}{dx_{i}dx_{k}}-\sum_{i}\frac{\partial (b_{i}v)}{\partial x_{i}}+cv=0\).Google Scholar - (1).See footnote on p. 576.Google Scholar

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© Verlag Harri Deutsch, Zürich 1973