Internal problem ID [3715]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 33
Problem number: 991.
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 {2 y^{2} \left (y^{\prime }\right )^{2}+2 x y^{\prime } y-1+x^{2}+y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.16 (sec). Leaf size: 133
dsolve(2*y(x)^2*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)-1+x^2+y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\sqrt {-2 x^{2}+4}}{2} \\ y \relax (x ) = \frac {\sqrt {-2 x^{2}+4}}{2} \\ y \relax (x ) = \sqrt {-2 \sqrt {1-c_{1}}\, x +2 c_{1}-x^{2}-1} \\ y \relax (x ) = \sqrt {2 \sqrt {1-c_{1}}\, x +2 c_{1}-x^{2}-1} \\ y \relax (x ) = -\sqrt {-2 \sqrt {1-c_{1}}\, x +2 c_{1}-x^{2}-1} \\ y \relax (x ) = -\sqrt {2 \sqrt {1-c_{1}}\, x +2 c_{1}-x^{2}-1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.541 (sec). Leaf size: 57
DSolve[2 y[x]^2 (y'[x])^2 +2 x y[x] y'[x]-1+x^2+y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-x^2+c_1 x+1-\frac {c_1{}^2}{2}} \\ y(x)\to \sqrt {-x^2+c_1 x+1-\frac {c_1{}^2}{2}} \\ \end{align*}