Internal problem ID [9285]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1707 (book 6.116).
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^{\prime }\relax (x )-f^{\prime \prime }\relax (x ) y+f \relax (x ) y^{3}-y^{4}=0} \end {gather*}
✗ Solution by Maple
dsolve(diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2+diff(f(x),x)*diff(y(x),x)-diff(diff(f(x),x),x)*y(x)+f(x)*y(x)^3-y(x)^4=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.272 (sec). Leaf size: 235
DSolve[f[x]*y[x]^3 - y[x]^4 + Derivative[1][f][x]*y'[x] - y'[x]^2 - y[x]*Derivative[2][f][x] + 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 (y(K[3])^2-f(K[3]) y(K[3])+\left (c_1+\int _1^{K[3]}\frac {f(K[1]) y(K[1])^3-\left (y(K[1])^3+f''(K[1])\right ) y(K[1])+f'(K[1]) y'(K[1])}{y(K[1])^2}dK[1]\right ){}^2\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])^3-\left (y(K[1])^3+f''(K[1])\right ) y(K[1])+f'(K[1]) y'(K[1])}{y(K[1])^2}dK[1]\right )}dK[3]\right )}{\int _1^x\frac {f(K[1]) y(K[1])^3-\left (y(K[1])^3+f''(K[1])\right ) y(K[1])+f'(K[1]) y'(K[1])}{y(K[1])^2}dK[1]+c_1} \\ \end{align*}