Internal problem ID [8928]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1349.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (x^{2}+1\right ) y^{\prime }}{x^{3}}+\frac {y}{x^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 73
dsolve(diff(diff(y(x),x),x) = -(x^2+1)/x^3*diff(y(x),x)-1/x^4*y(x),y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{\frac {1}{4 x^{2}}} \left (\left (2 x^{2}-1\right ) \BesselI \left (0, \frac {1}{4 x^{2}}\right )+\BesselI \left (1, \frac {1}{4 x^{2}}\right )\right )}{x^{2}}+\frac {c_{2} {\mathrm e}^{\frac {1}{4 x^{2}}} \left (\left (2 x^{2}-1\right ) \BesselK \left (0, -\frac {1}{4 x^{2}}\right )+\BesselK \left (1, -\frac {1}{4 x^{2}}\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.068 (sec). Leaf size: 73
DSolve[y''[x] == -(y[x]/x^4) - ((1 + x^2)*y'[x])/x^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 G_{1,2}^{2,0}\left (-\frac {1}{2 x^2}| {c} \frac {3}{2} \\ 0,0 \\ \\ \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 )}{2 x^2} \\ \end{align*}