Internal problem ID [9300]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1722 (book 6.131).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [NONE]
Solve \begin {gather*} \boxed {y^{\prime \prime } y-\frac {\left (a -1\right ) \left (y^{\prime }\right )^{2}}{a}-y^{2} y^{\prime } f \relax (x )+\frac {a f \relax (x )^{2} y^{4}}{\left (a +2\right )^{2}}-\frac {a f^{\prime }\relax (x ) y^{3}}{a +2}=0} \end {gather*}
✗ 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: 19.791 (sec). Leaf size: 41
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*y'[x] - ((-1 + a)*y'[x]^2)/a + y[x]*y''[x] == 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} \\ \end{align*}