29.33.28 problem 991

Internal problem ID [5567]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 33
Problem number : 991
Date solved : Monday, January 27, 2025 at 12:04:15 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} 2 y^{2} {y^{\prime }}^{2}+2 x y y^{\prime }-1+x^{2}+y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.183 (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: 0.548 (sec). Leaf size: 57

DSolve[2 y[x]^2 (D[y[x],x])^2 +2 x y[x] D[y[x],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*}