Internal problem ID [8786]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1208.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+x^{3} y^{\prime }+\left (x^{2}-2\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 40
dsolve(x^2*diff(diff(y(x),x),x)+x^3*diff(y(x),x)+(x^2-2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1}}{x}+\frac {c_{2} \left (-\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )+2 x \,{\mathrm e}^{-\frac {x^{2}}{2}}\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.052 (sec). Leaf size: 49
DSolve[(-2 + x^2)*y[x] + x^3*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {2 \pi } c_2 \text {erf}\left (\frac {x}{\sqrt {2}}\right )-2 c_2 e^{-\frac {x^2}{2}} x+2 c_1}{2 x} \\ \end{align*}