29.23.11 problem 642

Internal problem ID [5233]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 642
Date solved : Sunday, March 30, 2025 at 07:06:05 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} x \left (a -x^{2}-y^{2}\right ) y^{\prime }+\left (a +x^{2}+y^{2}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 185

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

\begin{align*} \frac {y^{2} \left (-x^{2}+a \right )}{a -x^{2}-y^{2}} &= \frac {x^{2}}{2}-\frac {a}{2} \\ \frac {y^{2} \left (-x^{2}+a \right )}{a -x^{2}-y^{2}} &= -\frac {\sqrt {x^{2}-a}\, x}{\sqrt {\frac {-c_1 \,x^{2}+c_1 a -4 a}{-x^{2}+a}}}+\frac {x^{2}}{2}-\frac {a}{2} \\ \frac {y^{2} \left (-x^{2}+a \right )}{a -x^{2}-y^{2}} &= \frac {\sqrt {x^{2}-a}\, x}{\sqrt {\frac {-c_1 \,x^{2}+c_1 a -4 a}{-x^{2}+a}}}+\frac {x^{2}}{2}-\frac {a}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.937 (sec). Leaf size: 65

ode=x(a-x^2-y[x]^2)D[y[x],x]+(a+x^2+y[x]^2)y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{2} \left (c_1 x-\sqrt {-4 a+\left (4+c_1{}^2\right ) x^2}\right ) \\ y(x)\to \frac {1}{2} \left (\sqrt {-4 a+\left (4+c_1{}^2\right ) x^2}+c_1 x\right ) \\ \end{align*}

✗ Sympy

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

Timed Out