Internal problem ID [9342]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1763.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {x y y^{\prime \prime }+2 x \left (y^{\prime }\right )^{2}+y^{\prime } y a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.064 (sec). Leaf size: 238
dsolve(x*y(x)*diff(diff(y(x),x),x)+2*x*diff(y(x),x)^2+a*y(x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \frac {x^{-a} \left (\left (3 c_{2} x^{a} a -3 c_{2} x^{a}-3 x c_{1}\right ) \left (a -1\right )^{2} x^{2 a}\right )^{\frac {1}{3}}}{a -1} \\ y \relax (x ) = -\frac {x^{-a} \left (\left (3 c_{2} x^{a} a -3 c_{2} x^{a}-3 x c_{1}\right ) \left (a -1\right )^{2} x^{2 a}\right )^{\frac {1}{3}}}{2 \left (a -1\right )}-\frac {i \sqrt {3}\, x^{-a} \left (\left (3 c_{2} x^{a} a -3 c_{2} x^{a}-3 x c_{1}\right ) \left (a -1\right )^{2} x^{2 a}\right )^{\frac {1}{3}}}{2 \left (a -1\right )} \\ y \relax (x ) = -\frac {x^{-a} \left (\left (3 c_{2} x^{a} a -3 c_{2} x^{a}-3 x c_{1}\right ) \left (a -1\right )^{2} x^{2 a}\right )^{\frac {1}{3}}}{2 \left (a -1\right )}+\frac {i \sqrt {3}\, x^{-a} \left (\left (3 c_{2} x^{a} a -3 c_{2} x^{a}-3 x c_{1}\right ) \left (a -1\right )^{2} x^{2 a}\right )^{\frac {1}{3}}}{2 a -2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.174 (sec). Leaf size: 29
DSolve[a*y[x]*y'[x] + 2*x*y'[x]^2 + x*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 \sqrt [3]{3 x^{1-a}-a c_1+c_1} \\ \end{align*}