23.1.619 problem 613

Internal problem ID [5226]
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 : 613
Date solved : Sunday, October 12, 2025 at 01:24:00 AM
CAS classification : [_rational]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 110

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

\begin{align*} y &= \frac {x^{2}-c_1 x +\sqrt {x^{4}-2 c_1 \,x^{3}+\left (c_1^{2}-2\right ) x^{2}+\left (2 c_1 +4\right ) x -4 c_1 +1}-1}{-2 x +2 c_1} \\ y &= \frac {-x^{2}+c_1 x +\sqrt {x^{4}-2 c_1 \,x^{3}+\left (c_1^{2}-2\right ) x^{2}+\left (2 c_1 +4\right ) x -4 c_1 +1}+1}{2 x -2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 1.427 (sec). Leaf size: 146

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

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

✗ Sympy

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

Timed Out