29.21.4 problem 580

Internal problem ID [5174]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 21
Problem number : 580
Date solved : Sunday, March 30, 2025 at 06:47:45 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: 0.883 (sec). Leaf size: 115

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

✓ Sympy. Time used: 0.790 (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)
 

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