Internal problem ID [11004]
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: 170.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (\lambda \left (c +a \right ) x^{2}+\left (c -a \right ) x +2 b \lambda \right ) y^{\prime }+\lambda ^{2} \left (c \,x^{2}+b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 1571
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: 5.408 (sec). Leaf size: 104
DSolve[(a*x^2+b)*y''[x]+(\[Lambda]*(c+a)*x^2+(c-a)*x+2*b*\[Lambda])*y'[x]+\[Lambda]^2*(c*x^2+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{\lambda (-x)} (\lambda x+1) \left (c_2 \int _1^x\frac {\exp \left (\frac {(a-c) \lambda \left (\sqrt {a} K[1]-\sqrt {b} \arctan \left (\frac {\sqrt {a} K[1]}{\sqrt {b}}\right )\right )}{a^{3/2}}\right ) \left (a K[1]^2+b\right )^{\frac {a-c}{2 a}}}{(\lambda K[1]+1)^2}dK[1]+c_1\right ) \]