61.30.22 problem 170

Internal problem ID [12670]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-5
Problem number : 170
Date solved : Tuesday, January 28, 2025 at 08:10:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 12.128 (sec). Leaf size: 1377

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

\[ \text {Expression too large to display} \]

✓ Solution by Mathematica

Time used: 2.533 (sec). Leaf size: 96

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

\[ y(x)\to e^{-\lambda ^2 \int \frac {x}{\lambda x+1} \, dx} \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}\frac {K[1] \left (a \lambda ^2 K[1]^2-c (\lambda K[1]+1)^2+a\right )-2 b \lambda }{(\lambda K[1]+1) \left (a K[1]^2+b\right )}dK[1]\right )dK[2]+c_1\right ) \]