Internal problem ID [8793]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1213.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x^{2}+\left (x^{3}+1\right ) x y^{\prime }-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 59
dsolve(x^2*diff(diff(y(x),x),x)+(x^3+1)*x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {x^{3}}{6}} x^{\frac {3}{2}} \left (\BesselI \left (-\frac {1}{6}, \frac {x^{3}}{6}\right )+\BesselI \left (\frac {5}{6}, \frac {x^{3}}{6}\right )\right )+c_{2} {\mathrm e}^{-\frac {x^{3}}{6}} x^{\frac {3}{2}} \left (\BesselK \left (\frac {5}{6}, \frac {x^{3}}{6}\right )-\BesselK \left (\frac {1}{6}, \frac {x^{3}}{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.062 (sec). Leaf size: 54
DSolve[-y[x] + x*(1 + x^3)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{3} c_1 \, _1F_1\left (-\frac {1}{3};\frac {1}{3};-\frac {x^3}{3}\right )}{x}+\frac {c_2 x \, _1F_1\left (\frac {1}{3};\frac {5}{3};-\frac {x^3}{3}\right )}{\sqrt [3]{3}} \\ \end{align*}