60.7.115 problem 1706 (book 6.115)

Internal problem ID [11704]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1706 (book 6.115)
Date solved : Monday, January 27, 2025 at 11:30:25 PM
CAS classification : [NONE]

\begin{align*} y^{\prime \prime } y-{y^{\prime }}^{2}+f \left (x \right ) y^{\prime }-f^{\prime }\left (x \right ) y-y^{3}&=0 \end{align*}

✗ 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: 36.227 (sec). Leaf size: 197

DSolve[-y[x]^3 - y[x]*Derivative[1][f][x] + f[x]*D[y[x],x] - D[y[x],x]^2 + y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} 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} \\ y(x)\to 0 \\ \end{align*}