Internal problem ID [10086]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1764 (book 6.173).
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]]
\[ \boxed {x y y^{\prime \prime }+2 x {y^{\prime }}^{2}+a y^{\prime } y=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 164
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 \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {3^{\frac {1}{3}} \left (\left (a -1\right )^{2} \left (-x^{1+2 a} c_{1} +c_{2} x^{3 a} \left (a -1\right )\right )\right )^{\frac {1}{3}} x^{-a}}{a -1} \\ y \left (x \right ) &= -\frac {\left (\left (a -1\right )^{2} \left (-x^{1+2 a} c_{1} +c_{2} x^{3 a} \left (a -1\right )\right )\right )^{\frac {1}{3}} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) x^{-a}}{2 a -2} \\ y \left (x \right ) &= \frac {\left (\left (a -1\right )^{2} \left (-x^{1+2 a} c_{1} +c_{2} x^{3 a} \left (a -1\right )\right )\right )^{\frac {1}{3}} \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) x^{-a}}{2 a -2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.155 (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]
\[ y(x)\to c_2 \sqrt [3]{3 x^{1-a}-a c_1+c_1} \]