Internal problem ID [11039]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-6 Equation of form
\((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 205.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (x^{3} a +b \,x^{2}+x c +d \right ) y^{\prime \prime }-\left (-\lambda ^{2}+x^{2}\right ) y^{\prime }+\left (\lambda +x \right ) y=0} \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 87
dsolve((a*x^3+b*x^2+c*x+d)*diff(y(x),x$2)-(x^2-lambda^2)*diff(y(x),x)+(x+lambda)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (-\lambda +x \right )+c_{2} \left (\lambda -x \right ) \left (\int {\mathrm e}^{\int \frac {\left (-2 a +1\right ) x^{3}+\left (-2 b -\lambda \right ) x^{2}+\left (-\lambda ^{2}-2 c \right ) x +\lambda ^{3}-2 d}{\left (a \,x^{3}+b \,x^{2}+c x +d \right ) \left (-\lambda +x \right )}d x}d x \right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(a*x^3+b*x^2+c*x+d)*y''[x]-(x^2-\[Lambda]^2)*y'[x]+(x+\[Lambda])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Timed out