61.26.5 problem 5

Internal problem ID [12505]
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
Date solved : Tuesday, January 28, 2025 at 03:18:59 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+a^{3} x \left (-a x +2\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 26

dsolve(diff(y(x),x$2)+a^3*x*(2-a*x)*y(x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {a x \left (a x -2\right )}{2}} \left (c_{1} +\operatorname {erf}\left (i a x -i\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.370 (sec). Leaf size: 44

DSolve[D[y[x],{x,2}]+a^3*x*(2-a*x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-\frac {1}{2} a x (a x-2)} \left (c_2 \int _1^xe^{a K[1] (a K[1]-2)}dK[1]+c_1\right ) \]