Internal problem ID [8738]
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]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x^{2}+a y^{\prime }-\left (x^{2} b^{2}+a b \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 180
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)
\[ y \relax (x ) = c_{1} {\mathrm e}^{\frac {-b \,x^{2}+a}{x}} \mathit {HD}\left (4 \sqrt {2}\, \sqrt {b a}, -1-4 \sqrt {2}\, \sqrt {b a}, 8 \sqrt {2}\, \sqrt {b a}, -4 \sqrt {2}\, \sqrt {b a}+1, \frac {\sqrt {2}\, \sqrt {b a}\, x -a}{\sqrt {2}\, \sqrt {b a}\, x +a}\right ) \sqrt {x}+c_{2} {\mathrm e}^{b x} \mathit {HD}\left (-4 \sqrt {2}\, \sqrt {b a}, -1-4 \sqrt {2}\, \sqrt {b a}, 8 \sqrt {2}\, \sqrt {b a}, -4 \sqrt {2}\, \sqrt {b a}+1, \frac {\sqrt {2}\, \sqrt {b a}\, x -a}{\sqrt {2}\, \sqrt {b a}\, x +a}\right ) \sqrt {x} \]
✓ Solution by Mathematica
Time used: 0.125 (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]
\begin{align*} 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 ) \\ \end{align*}