MATHEMATICS

A. P. PRUDNIKOV

ON FUNCTIONS SATISFYING THE DIFFERENTIAL EQUATION \(x^2y''' + 3xy'' + y' + x^2y = 0\)

(Presented by Academician A. A. Dorodnitsyn on 19 XII 1961)

In the present note, by an operational method we solve the linear ordinary differential equation with variable coefficients

\[ x^2y''' + 3xy'' + y' + x^2y = 0. \tag{1} \]

After the change of variable by the formula \(x = 3\sqrt[3]{t}\), equation (1) is reduced to the form

\[ \frac{d}{dt}\,t\,\frac{d}{dt}\,t\,\frac{d}{dt}\,y + y = 0. \tag{2} \]

This equation is of importance in constructing an operational calculus for the operator
\[ T=\frac{d}{dt}\,t\,\frac{d}{dt}\,t\,\frac{d}{dt} \]
\((^1)\).

To obtain three linearly independent solutions of the latter equation by the operational method, it is expedient to consider the equation

\[ t^3y''' + 3t^2y'' + ty' + ty = 0, \tag{3} \]

which is obtained from (2) by multiplying by \(t\).

Let

\[ \varphi(p)=\int_0^\infty e^{-pt}y(t)\,dt. \]

Then equation (3) is reduced to the operational equation

\[ p^3\varphi''' + 6p^2\varphi'' + (7p+1)\varphi' + \varphi = 0. \tag{4} \]

After the change of variable by the formula \(z=\frac{1}{p}\), equation (4) becomes

\[ z^3\varphi''' + z(z+1)\varphi' - \varphi = 0. \tag{5} \]

Equation (5) is a special case of an equation depending on a parameter \(\nu\):

\[ z^3\varphi''' + \left[z(z+1)-\left(\frac{\nu}{2}\right)^2 z\right]\varphi' -\left[1-\left(\frac{\nu}{2}\right)\right]\varphi = 0, \tag{6} \]

when this parameter is equal to zero. If the parameter \(\nu\) is not an integer, then the functions

\[ \widetilde{\varphi}_1(z,\nu)=zJ_\nu(2\sqrt{z}); \tag{7} \]

\[ \widetilde{\varphi}_2(z,\nu)=zY_\nu(2\sqrt{z}); \tag{8} \]

\[ \widetilde{\varphi}_3(z,\nu)=z\Pi_\nu(2\sqrt{z}), \tag{9} \]

where

\[ J_\nu(z)=\sum_{m=0}^{\infty}\frac{(-1)^m(z/2)^{\nu+2m}}{m!\Gamma(\nu+m+1)},\qquad Y_\nu(z)=\frac{\cos(\pi\nu)J_\nu(z)-J_{-\nu}(z)}{\sin(\pi\nu)} \]

are Bessel functions,

\[ \Pi_\nu(z)=\cos\left(\frac{\pi\nu}{2}\right) \sum_{m=0}^{\infty} \frac{(-1)^m(z/2)^{2m}} {\Gamma(m+1+\nu/2)\Gamma(m+1-\nu/2)} \]

is a Poisson function, form a fundamental system of solutions of equation (6). For \(\nu=0\) this no longer holds, since as \(\nu\to0\) the Poisson function \(\Pi_\nu(z)\) degenerates into the Bessel function \(J_\nu(z)\), and the triple of solutions (7), (8), (9) of equation (6) degenerates into a pair of linearly independent solutions of equation (5)

\[ \varphi_1(z)=\lim_{\nu\to0}\widetilde{\varphi}_1(z,\nu) =\lim_{\nu\to0}\widetilde{\varphi}_3(z,\nu) =zJ_0(2\sqrt z); \tag{10} \]

\[ \varphi_2(z)=\lim_{\nu\to0}\widetilde{\varphi}_2(z,\nu) =zY_0(2\sqrt z). \tag{11} \]

To find the third linearly independent solution of equation (5), let us consider

\[ \lim_{\nu\to0} \frac{\widetilde{\varphi}_1(z,\nu)-\widetilde{\varphi}_3(z,\nu)}{\nu} = \frac12\varphi_2(z). \]

We find

\[ \lim_{\nu\to0} \frac{ \dfrac{d}{d\nu}\{\widetilde{\varphi}_1(z,\nu)-\widetilde{\varphi}_3(z,\nu)\} -\dfrac12\varphi_2(z)} {\nu} = \frac{\pi^2}{4}\varphi_1(z)+\varphi_3(z), \]

where

\[ \varphi_3(z)=\frac14\sum_{m=0}^{\infty} \frac{(-1)^m z^{m+1}}{(m!)^2} \{\ln^2 z-4\psi(m+1)\ln z+4\psi^2(m+1)-2\psi'(m+1)\}, \]

\[ \psi(m+1)=\frac{d}{dm}\ln\Gamma(m+1). \]

The function \(\varphi_3(z)\) satisfies equation (5) and is linearly independent of \(\varphi_1(z)\) and \(\varphi_2(z)\). After passing from the images
\(\varphi_1\left(\frac1p\right)\), \(\varphi_2\left(\frac1p\right)\), \(\varphi_3\left(\frac1p\right)\) to the originals, we obtain, respectively, three linearly independent solutions of equation (3):

\[ y_1(t)=\sum_{m=0}^{\infty}\frac{(-1)^m}{(m!)^3}t^m =J_{0,0}^{(2)}(3\sqrt[3]{t}), \]

\[ y_2(t)=J_{0,0}^{(2)}(3\sqrt[3]{t})\ln t -3\sum_{m=0}^{\infty}\frac{(-1)^m}{(m!)^3}t^m\psi(m+1), \]

\[ y_3(t)=\frac14\sum_{m=0}^{\infty}\frac{(-1)^m}{(m!)^3}t^m \{\ln^2 t-6\psi(m+1)\ln t+9\psi^2(m+1)-3\psi'(m+1)\}. \]

Computing Center
Academy of Sciences of the USSR

Received
18 XII 1961

CITED LITERATURE

1. A. P. Prudnikov, DAN, 142, No. 4 (1962).