Internal problem ID [8775]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1195.
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 +3\right ) x y^{\prime }-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.026 (sec). Leaf size: 94
dsolve(x^2*diff(diff(y(x),x),x)+(x+3)*x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (\left (\sqrt {2}+x +1\right ) \BesselI \left (-\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right )+\BesselI \left (\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right ) \left (-\sqrt {2}+x +1\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{\sqrt {x}}+\frac {c_{2} \left (\left (\sqrt {2}+x +1\right ) \BesselK \left (-\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right )-\BesselK \left (\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right ) \left (-\sqrt {2}+x +1\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 63
DSolve[-y[x] + x*(3 + x)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-x} x^{\sqrt {2}-1} \left (c_1 \text {HypergeometricU}\left (2+\sqrt {2},1+2 \sqrt {2},x\right )+c_2 L_{-2-\sqrt {2}}^{2 \sqrt {2}}(x)\right ) \\ \end{align*}