Internal problem ID [8777]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1197.
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 (x^{2}-2 x \right ) y^{\prime }-\left (x +a \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.01 (sec). Leaf size: 49
dsolve(x^2*diff(diff(y(x),x),x)-(x^2-2*x)*diff(y(x),x)-(x+a)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{\frac {x}{2}} \BesselI \left (\frac {\sqrt {4 a +1}}{2}, \frac {x}{2}\right )}{\sqrt {x}}+\frac {c_{2} {\mathrm e}^{\frac {x}{2}} \BesselK \left (\frac {\sqrt {4 a +1}}{2}, \frac {x}{2}\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 67
DSolve[(-a - x)*y[x] - (-2*x + x^2)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{x/2} \left (c_1 J_{\frac {1}{2} \sqrt {4 a+1}}\left (-\frac {i x}{2}\right )+c_2 Y_{\frac {1}{2} \sqrt {4 a+1}}\left (-\frac {i x}{2}\right )\right )}{\sqrt {x}} \\ \end{align*}