Internal problem ID [9539]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1209.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{2}+2\right ) x y^{\prime }+\left (x^{2}-2\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 41
dsolve(x^2*diff(diff(y(x),x),x)+(x^2+2)*x*diff(y(x),x)+(x^2-2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-c_{2} \pi \,\operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right )+c_{1} \right ) {\mathrm e}^{-\frac {x^{2}}{2}}+i \sqrt {\pi }\, \sqrt {2}\, c_{2} x}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.032 (sec). Leaf size: 59
DSolve[(-2 + x^2)*y[x] + x*(2 + x^2)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-\frac {x^2}{2}} \left (2 \left (c_1 e^{\frac {x^2}{2}} x+c_2\right )-\sqrt {2 \pi } c_1 \text {erfi}\left (\frac {x}{\sqrt {2}}\right )\right )}{2 x^2} \]