Internal problem ID [10082]
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: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+a^{3} x \left (-a x +2\right ) y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 41
dsolve(diff(y(x),x$2)+a^3*x*(2-a*x)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {a x \left (a x -2\right )}{2}}+c_{2} {\mathrm e}^{-\frac {1}{2} a^{2} x^{2}+a x} \operatorname {erf}\left (i a x -i\right ) \]
✓ Solution by Mathematica
Time used: 0.081 (sec). Leaf size: 50
DSolve[y''[x]+a^3*x*(2-a*x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-\frac {1}{2} a x (a x-2)-1} \left (2 e a c_1-\sqrt {\pi } c_2 \text {erfi}(1-a x)\right )}{2 a} \\ \end{align*}