# Differential Equations

• I. N. Bronshtein
• K. A. Semendyayev

## 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

2. (1).
Complex values of functions are also taken into account.Google Scholar
3. (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
4. (2).
The general solution written in the form x2 = C1y contains also the line x = 0.Google Scholar
5. (1).
The coefficients ak are assumed to be real.Google Scholar
6. (1).
This method can also be applied in the case, when F(x) = Qp(x), i.e., when k = 0, and also in the cases, when F(x) = Qp(x) erx cos ωx or F(x) = Qp(x) erx sin ωx which corresponds to the value k = r ± iω. 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
7. (1).
This is an abbreviation for a determinant with elements aik.Google Scholar
8. (1).
The function f(p)/p is called the Laplace transform of φ(t).Google Scholar
9. (2).
$$\frac{d^{n}\varphi} {dt^{n}}$$ is assumed to satisfy the conditions introduced above under which the transform is defined.Google Scholar
10. (1).
11. (1).
For the Γ function, see p. 192.Google Scholar
12. (2).
This function is sometimes denoted by Nn(x).Google Scholar
13. (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
14. (1).
We shall assume that the interval (a, b) is finite. In the case of an infinite interval, the results change essentially.Google Scholar
15. (1).
In solving a system in this form, any of the variables xk for which Xk≠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
16. (1).
If at least two positive and at least two negative coefficients occur in the equation, then the equation is called ultrahyperbolic.Google Scholar
17. (1).
In this formula, the symbol x is used to denote the system of n variables x1, x2, …, xn 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
18. (1).
Δ is the Laplace’s operator with respect to n variables x1, x2, …, xn (see p. 566).Google Scholar
19. (2).
We use the notation exp x = ex.Google Scholar
20. (1).
Δ is the Laplace operator with respect to n variables x1, x2,…, xn (see p. 566).Google Scholar
21. (1).
It follows from the following discussion that the boundary conditions cannot be satisfied, when this constant is positive.Google Scholar
22. (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
23. (1).
See footnote on p. 576.Google Scholar