Internal problem ID [9364]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1786 (book 6.195).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
Solve \begin {gather*} \boxed {\left (y^{2}+x^{2}\right ) y^{\prime \prime }-2 \left (\left (y^{\prime }\right )^{2}+1\right ) \left (y^{\prime } x -y\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.219 (sec). Leaf size: 97
dsolve((y(x)^2+x^2)*diff(diff(y(x),x),x)-2*(diff(y(x),x)^2+1)*(x*diff(y(x),x)-y(x))=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -i x \\ y \relax (x ) = i x \\ y \relax (x ) = \frac {c_{1}+1-\sqrt {-4 c_{2}^{2} x^{2}+4 i c_{1} c_{2} x -4 i c_{2} x +c_{1}^{2}+2 c_{1}+1}}{2 c_{2}} \\ y \relax (x ) = \frac {c_{1}+1+\sqrt {-4 c_{2}^{2} x^{2}+4 i c_{1} c_{2} x -4 i c_{2} x +c_{1}^{2}+2 c_{1}+1}}{2 c_{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.249 (sec). Leaf size: 95
DSolve[-2*(-y[x] + x*y'[x])*(1 + y'[x]^2) + (x^2 + y[x]^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (-\sqrt {4 x \left (-x+e^{c_2}\right )+e^{2 c_2} \cot ^2(c_1)}-e^{c_2} \cot (c_1)\right ) \\ y(x)\to \frac {1}{2} \left (\sqrt {4 x \left (-x+e^{c_2}\right )+e^{2 c_2} \cot ^2(c_1)}-e^{c_2} \cot (c_1)\right ) \\ \end{align*}