Internal problem ID [8784]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1204.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+\left (a +2 b \right ) x^{2} y^{\prime }+\left (\left (a +b \right ) b \,x^{2}-2\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 37
dsolve(x^2*diff(diff(y(x),x),x)+(a+2*b)*x^2*diff(y(x),x)+((a+b)*b*x^2-2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{-b x} \left (a x -2\right )}{x}+\frac {c_{2} {\mathrm e}^{-x \left (a +b \right )} \left (a x +2\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 84
DSolve[(-2 + b*(a + b)*x^2)*y[x] + (a + 2*b)*x^2*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {a x^{3/2} e^{-\frac {1}{2} x (a+2 b)} \left (2 (i a c_2 x+2 c_1) \sinh \left (\frac {a x}{2}\right )-2 (a c_1 x+2 i c_2) \cosh \left (\frac {a x}{2}\right )\right )}{\sqrt {\pi } (-i a x)^{5/2}} \\ \end{align*}