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Method of integral transforms for solving some classes of differential equations. (Russian) Zbl 0659.34001

The author considers an integral transformation of the form \[ \phi_{\nu}(x) = \int^\infty_0 \cdots \int^\infty_0 f(x\prod t_ i^{1/2}) \quad \exp(-\sum t_ i) \prod t_ i^{\nu_ i} \quad dt_ i \]
\[ f(x) = \frac{1}{(2\pi i)^{r-1}} \int^{(0+)}_{-\infty} \cdots \int^{(0+)}_{-\infty} \phi_{\nu} (x\prod t_ i^{-1/r}) \quad \exp(\sigma t_ i) \prod t_ i^{-\nu_ i-1} \quad dt_ i \tag{\(*\)} \] and he applies it to solve linear ordinary differential equations. Under some conditions for a function f(x) he establishes the validity of (\(*\)). The transformation of differential operators are also considered. By using this transformation the author explicitly solves such equations as \[ y'' + (\frac{2\nu+1}{x} + ax)y' + by = f(x), \]
\[ (\frac{d^ r}{dx^ r} + \frac{b_ 1}{x} \frac{d^{r-1}}{dx^{r-1}} + \cdots + \frac{b_{r-1}}{x^{r-1}} \frac{d}{dx})y \to axy' + by = f(x). \] The direct transformation (\(*\)) reduces such equations to a linear ordinary equation of the first degree with a general solution of closed form. The inverse transformation leads to a general solution of the equation.
Reviewer: L.P.Lebedev

MSC:

34A05 Explicit solutions, first integrals of ordinary differential equations
34A30 Linear ordinary differential equations and systems
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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