Internal problem ID [10028]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1706 (book 6.115).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [NONE]
\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}+f \left (x \right ) y^{\prime }-f^{\prime }\left (x \right ) y-y^{3}=0} \]
✗ Solution by Maple
dsolve(diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2+f(x)*diff(y(x),x)-diff(f(x),x)*y(x)-y(x)^3=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 60.429 (sec). Leaf size: 192
DSolve[-y[x]^3 - y[x]*Derivative[1][f][x] + f[x]*y'[x] - y'[x]^2 + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {\exp \left (c_2-\int _1^x\frac {y(K[3])^3+\left (c_1+\int _1^{K[3]}-\frac {y(K[1])^3+f'(K[1]) y(K[1])-f(K[1]) y'(K[1])}{y(K[1])^2}dK[1]\right ){}^2 y(K[3])^2+f'(K[3]) y(K[3])-f(K[3]) y'(K[3])}{y(K[3])^2 \left (c_1+\int _1^{K[3]}-\frac {y(K[1])^3+f'(K[1]) y(K[1])-f(K[1]) y'(K[1])}{y(K[1])^2}dK[1]\right )}dK[3]\right )}{\int _1^x-\frac {y(K[1])^3+f'(K[1]) y(K[1])-f(K[1]) y'(K[1])}{y(K[1])^2}dK[1]+c_1} \]