Internal problem ID [11086]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-8. Other equations.
Problem number: 262.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+\left (\lambda ^{2}-x^{2}\right ) y^{\prime }+\left (\lambda +x \right ) y=0} \]
✓ Solution by Maple
Time used: 1.0 (sec). Leaf size: 76
dsolve((a*x^n+b*x^m+c)*diff(y(x),x$2)+(lambda^2-x^2)*diff(y(x),x)+(x+lambda)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = -\left (\left (\int {\mathrm e}^{\int \frac {\lambda ^{3}-x \,\lambda ^{2}-x^{2} \lambda +x^{3}-2 a \,x^{n}-2 b \,x^{m}-2 c}{\left (a \,x^{n}+b \,x^{m}+c \right ) \left (-\lambda +x \right )}d x}d x \right ) c_{1} +c_{2} \right ) \left (\lambda -x \right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(a*x^n+b*x^m+c)*y''[x]+(\[Lambda]^2-x^2)*y'[x]+(x+\[Lambda])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved