Internal problem ID [8070]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 490.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {y^{2} \left (y^{\prime }\right )^{2}-2 x y y^{\prime }+2 y^{2}-x^{2}+a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 145
dsolve(y(x)^2*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+2*y(x)^2-x^2+a = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\sqrt {4 x^{2}-2 a}}{2} \\ y \relax (x ) = \frac {\sqrt {4 x^{2}-2 a}}{2} \\ y \relax (x ) = \sqrt {-2 \sqrt {a +2 c_{1}}\, x -c_{1}-x^{2}-a} \\ y \relax (x ) = \sqrt {2 \sqrt {a +2 c_{1}}\, x -c_{1}-x^{2}-a} \\ y \relax (x ) = -\sqrt {-2 \sqrt {a +2 c_{1}}\, x -c_{1}-x^{2}-a} \\ y \relax (x ) = -\sqrt {2 \sqrt {a +2 c_{1}}\, x -c_{1}-x^{2}-a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.623 (sec). Leaf size: 63
DSolve[a - x^2 + 2*y[x]^2 - 2*x*y[x]*y'[x] + y[x]^2*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-\frac {a}{2}-x^2+4 c_1 x-2 c_1{}^2} \\ y(x)\to \sqrt {-\frac {a}{2}-x^2+4 c_1 x-2 c_1{}^2} \\ \end{align*}