23.1.649 problem 644

Internal problem ID [5256]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 644
Date solved : Tuesday, September 30, 2025 at 12:03:13 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 159

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

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

✓ Mathematica. Time used: 0.157 (sec). Leaf size: 192

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {-x^2-x}{2 (x+K[2]+1)^2}-\int _1^x\left (\frac {-2 K[2]-1}{2 (K[1]+K[2]+1)^2}+\frac {2 K[2]-1}{2 (K[1]+K[2]-1)^2}-\frac {-K[2]^2-K[2]}{(K[1]+K[2]+1)^3}-\frac {K[2]^2-K[2]}{(K[1]+K[2]-1)^3}\right )dK[1]+\frac {x^2-x}{2 (x+K[2]-1)^2}\right )dK[2]+\int _1^x\left (\frac {-y(x)^2-y(x)}{2 (K[1]+y(x)+1)^2}+\frac {y(x)^2-y(x)}{2 (K[1]+y(x)-1)^2}\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out