Internal problem ID [11096]
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: 0.125 (sec). Leaf size: 175
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)
\[ y \left (x \right ) = c_{1} \left (\int {\mathrm e}^{\int \frac {-2+\frac {\left (x +\lambda \right ) x^{2}}{a \,x^{n}+b \,x^{m}+c}-\frac {2 \left (x +\lambda \right ) x \lambda }{a \,x^{n}+b \,x^{m}+c}+\frac {\left (x +\lambda \right ) \lambda ^{2}}{a \,x^{n}+b \,x^{m}+c}}{-\lambda +x}d x}d x \right ) x -c_{1} \left (\int {\mathrm e}^{\int \frac {-2+\frac {\left (x +\lambda \right ) x^{2}}{a \,x^{n}+b \,x^{m}+c}-\frac {2 \left (x +\lambda \right ) x \lambda }{a \,x^{n}+b \,x^{m}+c}+\frac {\left (x +\lambda \right ) \lambda ^{2}}{a \,x^{n}+b \,x^{m}+c}}{-\lambda +x}d x}d x \right ) \lambda +c_{2} x -c_{2} \lambda \]
✗ 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