Том 71
№ 11

All Issues

On differential equations with functional parameters

Polishchuk Ye. M.

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The author investigates a differential equation of the form $$y^m = f(t, y, y',..., y^{m-1})\varphi(x_1(t), ...x_p(t))\quad (1)$$ where $t$ is an independent variable, $y$ is an unknown function, $x_1,...x_p$ are parameters depending on $t$. It is shown that under definite conditions in respect to $f$ and $\varphi$, if the function $\varphi$ satisfies the differential equation $$\sum_{k=1}^pa_k\frac{\partial^2\varphi}{\partial x^2_k} $$ then on varying the functions $x_1,...x_p$ the integral $Y[x_1,...x_p; t]$ of equation (1), considered as a functional of $x_1,...x_p$ satisfies the equation $\sum a_k\delta_k Y = 0$, where $\delta_k$ is a partial functional Laplace operator in respect to the function $x_k(t),\; k = 1, 2,.., p$. If equation (1) has the form $$y^m = f_1(t, y, y',..., y^{m-1}) + f_2(t, y, y',..., y^{m-1})$$ then on varying $x$, its integral $Y [x/t]$ satisfies the equation $\delta Y = 0$ (i. e. is a harmonic functional of $x$). As examples the Ricatti and Schwarz integrals are presented as discussed by Fantappie in his theory of analytical functionals.

Citation Example: Polishchuk Ye. M. On differential equations with functional parameters // Ukr. Mat. Zh. - 1963. - 15, № 1. - pp. 13-24.

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