Internal problem ID [8933]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1354.
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 (2 x^{2}-1\right ) y^{\prime }}{x^{3}}+\frac {2 y}{x^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 32
dsolve(diff(diff(y(x),x),x) = 1/x^3*(2*x^2-1)*diff(y(x),x)-2/x^4*y(x),y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (5 x^{2}-1\right )}{x^{2}}+c_{2} \hypergeom \left (\left [-\frac {5}{2}\right ], \left [-\frac {1}{2}\right ], \frac {1}{2 x^{2}}\right ) x^{3} \]
✓ Solution by Mathematica
Time used: 0.074 (sec). Leaf size: 73
DSolve[y''[x] == (-2*y[x])/x^4 + ((-1 + 2*x^2)*y'[x])/x^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\left (5 x^2-1\right ) \left (12 c_1-5 \sqrt {2 \pi } c_2 \text {Erfi}\left (\frac {1}{\sqrt {2} x}\right )\right )+10 c_2 e^{\frac {1}{2 x^2}} x \left (2 x^4+4 x^2-1\right )}{60 x^2} \\ \end{align*}