Internal problem ID [8934]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1355.
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^{3}-1\right ) y^{\prime }}{x \left (x^{3}+1\right )}-\frac {x y}{x^{3}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.023 (sec). Leaf size: 37
dsolve(diff(diff(y(x),x),x) = -(x^3-1)/x/(x^3+1)*diff(y(x),x)+x/(x^3+1)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{2} \left (x^{3}+1\right )^{\frac {1}{3}} \hypergeom \left (\left [\frac {2}{3}, \frac {4}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )+c_{2} \left (x^{3}+1\right )^{\frac {1}{3}} \]
✓ Solution by Mathematica
Time used: 4.06 (sec). Leaf size: 44
DSolve[y''[x] == (x*y[x])/(1 + x^3) - ((-1 + x^3)*y'[x])/(x*(1 + x^3)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \sqrt [3]{x^3+1} \left (c_2 x^2 \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {5}{3};-x^3\right )+2 c_1\right ) \\ \end{align*}