Internal problem ID [10087]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2 Equations Containing Power
Functions. page 213
Problem number: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\left (a \,x^{2}+b c x \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.051 (sec). Leaf size: 119
dsolve(diff(y(x),x$2)-(a*x^2+b*x*c)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [-\frac {b^{2} c^{2}-4 a^{\frac {3}{2}}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {\left (2 x a +c b \right )^{2}}{4 a^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {x \left (x a +c b \right )}{2 \sqrt {a}}}+c_{2} \left (2 x a +c b \right ) \hypergeom \left (\left [-\frac {b^{2} c^{2}-12 a^{\frac {3}{2}}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (2 x a +c b \right )^{2}}{4 a^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {x \left (x a +c b \right )}{2 \sqrt {a}}} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 92
DSolve[y''[x]-(a*x^2+b*x*c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 D_{-\frac {b^2 c^2}{8 a^{3/2}}-\frac {1}{2}}\left (\frac {i (b c+2 a x)}{\sqrt {2} a^{3/4}}\right )+c_1 D_{\frac {1}{8} \left (\frac {b^2 c^2}{a^{3/2}}-4\right )}\left (\frac {b c+2 a x}{\sqrt {2} a^{3/4}}\right ) \\ \end{align*}