Internal problem ID [9339]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1760.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {x y y^{\prime \prime }+x \left (y^{\prime }\right )^{2}+y^{\prime } y a +f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 106
dsolve(x*y(x)*diff(diff(y(x),x),x)+x*diff(y(x),x)^2+a*y(x)*diff(y(x),x)+f(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {-2 \left (a -1\right ) \left (-x^{-a +1} \left (\int \frac {x^{a} f \relax (x )}{x}d x \right )-x^{-a +1} c_{1}+\int f \relax (x )d x +c_{2}\right )}}{a -1} \\ y \relax (x ) = -\frac {\sqrt {-2 \left (a -1\right ) \left (-x^{-a +1} \left (\int \frac {x^{a} f \relax (x )}{x}d x \right )-x^{-a +1} c_{1}+\int f \relax (x )d x +c_{2}\right )}}{a -1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.065 (sec). Leaf size: 108
DSolve[f[x] + a*y[x]*y'[x] + x*y'[x]^2 + x*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {2} \sqrt {\int _1^x-K[2]^{-a} \left (c_1+\int _1^{K[2]}f(K[1]) K[1]^{a-1}dK[1]\right )dK[2]+c_2} \\ y(x)\to \sqrt {2} \sqrt {\int _1^x-K[2]^{-a} \left (c_1+\int _1^{K[2]}f(K[1]) K[1]^{a-1}dK[1]\right )dK[2]+c_2} \\ \end{align*}