Internal problem ID [4223]
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)]`]]
\[ \boxed {2 y^{2} {y^{\prime }}^{2}+2 x y y^{\prime }+y^{2}=-x^{2}+1} \]
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 103
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 \left (x \right ) = -\frac {\sqrt {-2 x^{2}+4}}{2} y \left (x \right ) = \frac {\sqrt {-2 x^{2}+4}}{2} y \left (x \right ) = \sqrt {\operatorname {RootOf}\left (-2 \ln \left (x \right )+2 \,\operatorname {arctanh}\left (\sqrt {-2 \textit {\_Z} -1}\right )-\ln \left (\textit {\_Z} +1\right )+2 c_{1} \right ) x^{2}+1} y \left (x \right ) = -\sqrt {\operatorname {RootOf}\left (-2 \ln \left (x \right )+2 \,\operatorname {arctanh}\left (\sqrt {-2 \textit {\_Z} -1}\right )-\ln \left (\textit {\_Z} +1\right )+2 c_{1} \right ) x^{2}+1} \end{align*}
✓ Solution by Mathematica
Time used: 0.576 (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*}