Internal problem ID [11014]
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: 180.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }-\left (-k^{2}+x^{2}\right ) y^{\prime }+\left (x +k \right ) y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 1802
dsolve((a*x^2+b*x+c)*diff(y(x),x$2)-(x^2-k^2)*diff(y(x),x)+(x+k)*y(x)=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 2.442 (sec). Leaf size: 119
DSolve[(a*x^2+b*x+c)*y''[x]-(x^2-k^2)*y'[x]+(x+k)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {(k-x) \left (c_2 \int _1^x\frac {\exp \left (\frac {\frac {\left (b^2-2 a \left (a k^2+c\right )\right ) \arctan \left (\frac {b+2 a K[1]}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+a K[1]}{a^2}\right ) (c+K[1] (b+a K[1]))^{-\frac {b}{2 a^2}}}{(k-K[1])^2}dK[1]+c_1\right )}{k} \]