Internal problem ID [10970]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-4 Equation of form
\(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 136.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (\left (-c +a \right ) x^{2}+b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 245
dsolve(x^2*diff(y(x),x$2)+(a*x^2+b)*diff(y(x),x)+c*((a-c)*x^2+b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {x}\, {\mathrm e}^{\frac {-c \,x^{2}+b}{x}} \operatorname {HeunD}\left (4 \sqrt {-b \left (-2 c +a \right )}, -4 \sqrt {-b \left (-2 c +a \right )}-1+\left (-4 a +8 c \right ) b , 8 \sqrt {-b \left (-2 c +a \right )}, -4 \sqrt {-b \left (-2 c +a \right )}+1+\left (-8 c +4 a \right ) b , \frac {\sqrt {-b \left (-2 c +a \right )}\, x -b}{\sqrt {-b \left (-2 c +a \right )}\, x +b}\right )+c_{2} \sqrt {x}\, {\mathrm e}^{-x \left (a -c \right )} \operatorname {HeunD}\left (-4 \sqrt {-b \left (-2 c +a \right )}, -4 \sqrt {-b \left (-2 c +a \right )}-1+\left (-4 a +8 c \right ) b , 8 \sqrt {-b \left (-2 c +a \right )}, -4 \sqrt {-b \left (-2 c +a \right )}+1+\left (-8 c +4 a \right ) b , \frac {\sqrt {-b \left (-2 c +a \right )}\, x -b}{\sqrt {-b \left (-2 c +a \right )}\, x +b}\right ) \]
✓ Solution by Mathematica
Time used: 1.026 (sec). Leaf size: 44
DSolve[x^2*y''[x]+(a*x^2+b)*y'[x]+c*((a-c)*x^2+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-c x} \left (c_2 \int _1^xe^{\frac {b}{K[1]}-a K[1]+2 c K[1]}dK[1]+c_1\right ) \]