Internal problem ID [10080]
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: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-\left (a^{2} x^{2}+a \right ) y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 29
dsolve(diff(y(x),x$2)-(a^2*x^2+a)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {a \,x^{2}}{2}}+c_{2} {\mathrm e}^{\frac {a \,x^{2}}{2}} \operatorname {erf}\left (\sqrt {a}\, x \right ) \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 36
DSolve[y''[x]-(a^2*x^2+a)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \operatorname {ParabolicCylinderD}\left (-1,\sqrt {2} \sqrt {a} x\right )+c_2 e^{\frac {a x^2}{2}} \\ \end{align*}