Internal problem ID [9683]
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]]
\[ \boxed {y^{\prime \prime }+\frac {\left (x^{3}-1\right ) y^{\prime }}{x \left (x^{3}+1\right )}-\frac {x y}{x^{3}+1}=0} \]
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 54
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)
\[ y \left (x \right ) = \frac {2 c_{1} x^{2} \operatorname {LegendreP}\left (-\frac {1}{3}, -\frac {2}{3}, \frac {-x^{3}+1}{x^{3}+1}\right ) \Gamma \left (\frac {2}{3}\right )}{3 \left (x^{3}+1\right )^{\frac {1}{3}} \left (-x^{3}\right )^{\frac {1}{3}}}+c_{2} \left (x^{3}+1\right )^{\frac {1}{3}} \]
✓ Solution by Mathematica
Time used: 8.037 (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]
\[ y(x)\to \frac {1}{2} \sqrt [3]{x^3+1} \left (c_2 x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},-x^3\right )+2 c_1\right ) \]