23.1.584 problem 577

Internal problem ID [5191]
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 : 577
Date solved : Tuesday, September 30, 2025 at 11:51:59 AM
CAS classification : [_rational, _Bernoulli]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 87

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

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

✓ Mathematica. Time used: 0.213 (sec). Leaf size: 90

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

\begin{align*} y(x)&\to -\sqrt {x^2-1} \sqrt {2 \int _1^x\frac {2 K[1]^2}{\left (K[1]^2-1\right )^2}dK[1]+c_1}\\ y(x)&\to \sqrt {x^2-1} \sqrt {2 \int _1^x\frac {2 K[1]^2}{\left (K[1]^2-1\right )^2}dK[1]+c_1} \end{align*}

✓ Sympy. Time used: 0.699 (sec). Leaf size: 87

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

\[ \left [ y{\left (x \right )} = - \sqrt {C_{1} x^{2} - C_{1} + x^{2} \log {\left (x - 1 \right )} - x^{2} \log {\left (x + 1 \right )} - 2 x - \log {\left (x - 1 \right )} + \log {\left (x + 1 \right )}}, \ y{\left (x \right )} = \sqrt {C_{1} x^{2} - C_{1} + x^{2} \log {\left (x - 1 \right )} - x^{2} \log {\left (x + 1 \right )} - 2 x - \log {\left (x - 1 \right )} + \log {\left (x + 1 \right )}}\right ] \]