Internal problem ID [9353]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1775 (book 6.184).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {a \,x^{2} y y^{\prime \prime }+b \,x^{2} \left (y^{\prime }\right )^{2}+c x y y^{\prime }+d y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 159
dsolve(a*x^2*y(x)*diff(diff(y(x),x),x)+b*x^2*diff(y(x),x)^2+c*x*y(x)*diff(y(x),x)+d*y(x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \left (\frac {a^{2}-2 c a -4 d a -4 b d +c^{2}}{\left (x^{\frac {\sqrt {a^{2}-2 c a -4 d a -4 b d +c^{2}}}{a}} c_{1} a +x^{\frac {\sqrt {a^{2}-2 c a -4 d a -4 b d +c^{2}}}{a}} c_{1} b -c_{2} a -c_{2} b \right )^{2}}\right )^{-\frac {a}{2 \left (a +b \right )}} x^{-\frac {\sqrt {a^{2}-2 c a -4 d a -4 b d +c^{2}}}{2 \left (a +b \right )}} x^{\frac {a}{2 b +2 a}} x^{-\frac {c}{2 \left (a +b \right )}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.789 (sec). Leaf size: 82
DSolve[d*y[x]^2 + c*x*y[x]*y'[x] + b*x^2*y'[x]^2 + a*x^2*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 x^{-\frac {a \left (\sqrt {\frac {(a-c)^2-4 d (a+b)}{a^2}}-1\right )+c}{2 (a+b)}} \left (x^{\sqrt {\frac {(a-c)^2-4 d (a+b)}{a^2}}}+c_1\right ){}^{\frac {a}{a+b}} \\ \end{align*}