23.1.590 problem 583

Internal problem ID [5197]
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 : 583
Date solved : Tuesday, September 30, 2025 at 11:54:30 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 42

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

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

✓ Mathematica. Time used: 3.904 (sec). Leaf size: 161

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

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

✓ Sympy. Time used: 0.488 (sec). Leaf size: 32

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

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