60.7.131 problem 1722 (book 6.131)

Internal problem ID [11720]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1722 (book 6.131)
Date solved : Monday, January 27, 2025 at 11:31:22 PM
CAS classification : [NONE]

\begin{align*} y^{\prime \prime } y-\frac {\left (a -1\right ) {y^{\prime }}^{2}}{a}-f \left (x \right ) y^{2} y^{\prime }+\frac {a f \left (x \right )^{2} y^{4}}{\left (a +2\right )^{2}}-\frac {a f^{\prime }\left (x \right ) y^{3}}{a +2}&=0 \end{align*}

✗ Solution by Maple

dsolve(diff(diff(y(x),x),x)*y(x)-(a-1)/a*diff(y(x),x)^2-f(x)*y(x)^2*diff(y(x),x)+a/(a+2)^2*f(x)^2*y(x)^4-a/(a+2)*diff(f(x),x)*y(x)^3=0,y(x), singsol=all)
 

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 48.973 (sec). Leaf size: 46

DSolve[(a*f[x]^2*y[x]^4)/(2 + a)^2 - (a*y[x]^3*Derivative[1][f][x])/(2 + a) - f[x]*y[x]^2*D[y[x],x] - ((-1 + a)*D[y[x],x]^2)/a + y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {(a+2) (x+c_1){}^a}{a \int _1^xf(K[5]) (c_1+K[5]){}^adK[5]+c_2} \\ y(x)\to 0 \\ \end{align*}