Internal problem ID [9362]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1783.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]]
Solve \begin {gather*} \boxed {\left (x +y^{2}\right ) y^{\prime \prime }-2 \left (x -y^{2}\right ) \left (y^{\prime }\right )^{3}+y^{\prime } \left (1+4 y^{\prime } y\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.194 (sec). Leaf size: 41
dsolve((x+y(x)^2)*diff(diff(y(x),x),x)-2*(x-y(x)^2)*diff(y(x),x)^3+diff(y(x),x)*(1+4*y(x)*diff(y(x),x))=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \sqrt {-x} \\ y \relax (x ) = -\sqrt {-x} \\ \frac {-y \relax (x ) c_{1}+\ln \left (x +y \relax (x )^{2}\right )+c_{2}+2}{y \relax (x )} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.84 (sec). Leaf size: 26
DSolve[-2*(x - y[x]^2)*y'[x]^3 + y'[x]*(1 + 4*y[x]*y'[x]) + (x + y[x]^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [x=-y(x)^2+c_2 e^{e^{-c_1} y(x)},y(x)\right ] \]