Internal problem ID [8888]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1309.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+y x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 75
dsolve(x^3*diff(diff(y(x),x),x)-(x^2-1)*diff(y(x),x)+x*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{\frac {1}{4 x^{2}}} \left (\left (2 x^{2}-1\right ) \BesselI \left (0, -\frac {1}{4 x^{2}}\right )-\BesselI \left (1, -\frac {1}{4 x^{2}}\right )\right )}{x}+\frac {c_{2} {\mathrm e}^{\frac {1}{4 x^{2}}} \left (\left (2 x^{2}-1\right ) \BesselK \left (0, -\frac {1}{4 x^{2}}\right )+\BesselK \left (1, -\frac {1}{4 x^{2}}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.064 (sec). Leaf size: 77
DSolve[x*y[x] - (-1 + x^2)*y'[x] + x^3*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 G_{1,2}^{2,0}\left (-\frac {1}{2 x^2}| {c} 1 \\ -\frac {1}{2},-\frac {1}{2} \\ \\ \right )+\frac {c_1 e^{\frac {1}{4 x^2}} \left (\left (2 x^2-1\right ) I_0\left (\frac {1}{4 x^2}\right )+I_1\left (\frac {1}{4 x^2}\right )\right )}{\sqrt {2} x} \\ \end{align*}