Internal problem ID [8086]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 508.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [`y=_G(x,y')`]
\[ \boxed {\left (y^{4}+y^{2} x^{2}-x^{2}\right ) {y^{\prime }}^{2}+2 y x y^{\prime }-y^{2}=0} \]
✓ Solution by Maple
Time used: 0.421 (sec). Leaf size: 64
dsolve((y(x)^4+x^2*y(x)^2-x^2)*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)-y(x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = -i x \\ y \left (x \right ) = i x \\ y \left (x \right ) = 0 \\ y \left (x \right ) = -\operatorname {arctanh}\left (\operatorname {RootOf}\left (\operatorname {arctanh}\left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}-2 \,\operatorname {arctanh}\left (\textit {\_Z} \right ) c_{1} \textit {\_Z}^{2}+c_{1}^{2} \textit {\_Z}^{2}+\textit {\_Z}^{2} x^{2}-x^{2}\right )\right )+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.505 (sec). Leaf size: 88
DSolve[-y[x]^2 + 2*x*y[x]*y'[x] + (-x^2 + x^2*y[x]^2 + y[x]^4)*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\sqrt {x^2+y(x)^2} y(x) \left (\log \left (\frac {x}{\sqrt {x^2+y(x)^2}}+1\right )-\log \left (1-\frac {x}{\sqrt {x^2+y(x)^2}}\right )\right )}{2 x^2 \sqrt {\frac {y(x)^2 \left (x^2+y(x)^2\right )}{x^4}}}+y(x)=c_1,y(x)\right ] \]