Internal problem ID [11003]
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 Equation of form
\((a x^2+b x+c) y''+f(x)y'+g(x)y=0\)
Problem number: 169.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (c \,x^{2}+d \right ) y^{\prime }+\lambda \left (\left (-\lambda a +c \right ) x^{2}+d -b \lambda \right ) y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 1074
dsolve((a*x^2+b)*diff(y(x),x$2)+(c*x^2+d)*diff(y(x),x)+lambda*((c-a*lambda)*x^2+d-b*lambda)*y(x)=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 2.859 (sec). Leaf size: 74
DSolve[(a*x^2+b)*y''[x]+(c*x^2+d)*y'[x]+\[Lambda]*((c-a*\[Lambda])*x^2+d-b*\[Lambda])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{\lambda (-x)} \left (c_2 \int _1^x\exp \left (\frac {(b c-a d) \arctan \left (\frac {\sqrt {a} K[1]}{\sqrt {b}}\right )}{a^{3/2} \sqrt {b}}+\left (2 \lambda -\frac {c}{a}\right ) K[1]\right )dK[1]+c_1\right ) \]