Internal problem ID [9363]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1784.
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 }-\left (\left (y^{\prime }\right )^{2}+1\right ) \left (y^{\prime } x -y\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.148 (sec). Leaf size: 95
dsolve((y(x)^2+x^2)*diff(diff(y(x),x),x)-(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 ) = \tan \left (\RootOf \left (\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) {\mathrm e}^{-\frac {2 i c_{1} \textit {\_Z}}{-1+c_{1}}} {\mathrm e}^{-\frac {2 c_{1} c_{2}}{-1+c_{1}}} x^{-\frac {2 c_{1}}{-1+c_{1}}} {\mathrm e}^{-\frac {2 i \textit {\_Z}}{-1+c_{1}}} {\mathrm e}^{\frac {2 c_{2}}{-1+c_{1}}} x^{\frac {2}{-1+c_{1}}}-1\right )\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.208 (sec). Leaf size: 74
DSolve[(y[x] - x*y'[x])*(1 + y'[x]^2) + (x^2 + y[x]^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{2} \left (\log \left (1-\frac {i y(x)}{x}\right )+\log \left (1+\frac {i y(x)}{x}\right )+i \cot (c_1) \left (\log \left (1-\frac {i y(x)}{x}\right )-\log \left (1+\frac {i y(x)}{x}\right )\right )\right )=-\log (x)+c_2,y(x)\right ] \]