Internal problem ID [9284]
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]
Solve \begin {gather*} \boxed {y^{\prime \prime } y-\left (y^{\prime }\right )^{2}+y^{\prime } f \relax (x )-y f^{\prime }\relax (x )-y^{3}=0} \end {gather*}
✗ 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: 0.179 (sec). Leaf size: 190
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]
\begin{align*} y(x)\to -\frac {\exp \left (c_2-\int _1^x\frac {y(K[3]) \left (y(K[3]) \left (\left (c_1+\int _1^{K[3]}\frac {f(K[1]) y'(K[1])-y(K[1]) \left (y(K[1])^2+f'(K[1])\right )}{y(K[1])^2}dK[1]\right ){}^2+y(K[3])\right )+f'(K[3])\right )-f(K[3]) y'(K[3])}{y(K[3])^2 \left (c_1+\int _1^{K[3]}\frac {f(K[1]) y'(K[1])-y(K[1]) \left (y(K[1])^2+f'(K[1])\right )}{y(K[1])^2}dK[1]\right )}dK[3]\right )}{\int _1^x\frac {f(K[1]) y'(K[1])-y(K[1]) \left (y(K[1])^2+f'(K[1])\right )}{y(K[1])^2}dK[1]+c_1} \\ \end{align*}