Internal problem ID [10085]
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: 4.
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 \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.096 (sec). Leaf size: 45
dsolve(diff(y(x),x$2)-(a*x^2+b)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \WhittakerM \left (-\frac {b}{4 \sqrt {a}}, \frac {1}{4}, \sqrt {a}\, x^{2}\right )}{\sqrt {x}}+\frac {c_{2} \WhittakerW \left (-\frac {b}{4 \sqrt {a}}, \frac {1}{4}, \sqrt {a}\, x^{2}\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 68
DSolve[y''[x]-(a*x^2+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 D_{-\frac {b}{2 \sqrt {a}}-\frac {1}{2}}\left (\sqrt {2} \sqrt [4]{a} x\right )+c_2 D_{\frac {1}{2} \left (\frac {b}{\sqrt {a}}-1\right )}\left (i \sqrt {2} \sqrt [4]{a} x\right ) \\ \end{align*}