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 : Sunday, March 30, 2025 at 08:59:14 AM
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*}

✓ Maple. Time used: 0.141 (sec). Leaf size: 103

ode:=2*y(x)^2*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)-1+x^2+y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {-2 x^{2}+4}}{2} \\ y &= \frac {\sqrt {-2 x^{2}+4}}{2} \\ y &= \sqrt {\operatorname {RootOf}\left (-2 \ln \left (x \right )+2 \,\operatorname {arctanh}\left (\sqrt {-1-2 \textit {\_Z}}\right )-\ln \left (\textit {\_Z} +1\right )+2 c_1 \right ) x^{2}+1} \\ y &= -\sqrt {\operatorname {RootOf}\left (-2 \ln \left (x \right )+2 \,\operatorname {arctanh}\left (\sqrt {-1-2 \textit {\_Z}}\right )-\ln \left (\textit {\_Z} +1\right )+2 c_1 \right ) x^{2}+1} \\ \end{align*}

✓ Mathematica. Time used: 0.548 (sec). Leaf size: 57

ode=2 y[x]^2 (D[y[x],x])^2 +2 x y[x] D[y[x],x]-1+x^2+y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2 + 2*x*y(x)*Derivative(y(x), x) + 2*y(x)**2*Derivative(y(x), x)**2 + y(x)**2 - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out