61.27.5 problem 15

Internal problem ID [12515]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 15
Date solved : Tuesday, January 28, 2025 at 03:19:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 30

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

\[ y = \left (\left (\int {\mathrm e}^{-a x +\frac {2}{3} b \,x^{3}}d x \right ) c_{1} +c_{2} \right ) {\mathrm e}^{-\frac {b \,x^{3}}{3}} \]

✓ Solution by Mathematica

Time used: 0.967 (sec). Leaf size: 46

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

\[ y(x)\to e^{-\frac {b x^3}{3}} \left (c_2 \int _1^xe^{\frac {2}{3} b K[1]^3-a K[1]}dK[1]+c_1\right ) \]