Internal problem ID [10974]
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: 140.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{2}+\left (b a -1\right ) x +b \right ) y^{\prime }+a^{2} b x y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 298
dsolve(x^2*diff(y(x),x$2)+(a*x^2+(a*b-1)*x+b)*diff(y(x),x)+a^2*b*x*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{1-\frac {a b}{2}} {\mathrm e}^{\frac {-a \,x^{2}+b}{x}} \operatorname {HeunD}\left (4 \sqrt {a b}, \frac {\left (-2 a b -2\right ) \left (a b \right )^{\frac {3}{2}}-b a \left (\left (-2 a b +6\right ) \sqrt {a b}+\left (a b -2\right )^{2}\right )}{a b}, -8 \sqrt {a b}\, \left (a b -1\right ), \frac {\left (-2 a b -2\right ) \left (a b \right )^{\frac {3}{2}}+b a \left (\left (2 a b -6\right ) \sqrt {a b}+\left (a b -2\right )^{2}\right )}{a b}, \frac {\sqrt {a b}\, x -b}{\sqrt {a b}\, x +b}\right )+c_{2} x^{1-\frac {a b}{2}} \operatorname {HeunD}\left (-4 \sqrt {a b}, \frac {\left (-2 a b -2\right ) \left (a b \right )^{\frac {3}{2}}-b a \left (\left (-2 a b +6\right ) \sqrt {a b}+\left (a b -2\right )^{2}\right )}{a b}, -8 \sqrt {a b}\, \left (a b -1\right ), \frac {\left (-2 a b -2\right ) \left (a b \right )^{\frac {3}{2}}+b a \left (\left (2 a b -6\right ) \sqrt {a b}+\left (a b -2\right )^{2}\right )}{a b}, \frac {\sqrt {a b}\, x -b}{\sqrt {a b}\, x +b}\right ) \]
✓ Solution by Mathematica
Time used: 4.002 (sec). Leaf size: 67
DSolve[x^2*y''[x]+(a*x^2+(a*b-1)*x+b)*y'[x]+a^2*b*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-a x} (a x+1) \left (c_2 \int _1^x\frac {a^2 e^{\frac {b}{K[1]}+a K[1]} K[1]^{1-a b}}{(a K[1]+1)^2}dK[1]+c_1\right )}{a} \]