Internal problem ID [9488]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1158.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+a y^{\prime }-\left (b^{2} x^{2}+a b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.5 (sec). Leaf size: 178
dsolve(x^2*diff(diff(y(x),x),x)+a*diff(y(x),x)-(b^2*x^2+a*b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \sqrt {x}\, \left (\operatorname {HeunD}\left (4 \sqrt {2}\, \sqrt {a b}, -1-4 \sqrt {2}\, \sqrt {a b}, 8 \sqrt {2}\, \sqrt {a b}, -4 \sqrt {2}\, \sqrt {a b}+1, \frac {\sqrt {2}\, \sqrt {a b}\, x -a}{\sqrt {2}\, \sqrt {a b}\, x +a}\right ) {\mathrm e}^{\frac {-b \,x^{2}+a}{x}} c_{1} +{\mathrm e}^{b x} \operatorname {HeunD}\left (-4 \sqrt {2}\, \sqrt {a b}, -1-4 \sqrt {2}\, \sqrt {a b}, 8 \sqrt {2}\, \sqrt {a b}, -4 \sqrt {2}\, \sqrt {a b}+1, \frac {\sqrt {2}\, \sqrt {a b}\, x -a}{\sqrt {2}\, \sqrt {a b}\, x +a}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.576 (sec). Leaf size: 38
DSolve[(-(a*b) - b^2*x^2)*y[x] + a*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{b x} \left (c_2 \int _1^xe^{\frac {a}{K[1]}-2 b K[1]}dK[1]+c_1\right ) \]