83.2.3 problem 1 (c)

Internal problem ID [21914]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter III. First order differential equations of the first degree. Ex. III at page 35
Problem number : 1 (c)
Date solved : Thursday, October 02, 2025 at 08:09:28 PM
CAS classification : [_separable]

\begin{align*} 1+y+y^{2}+x \left (x^{2}-4\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 33

ode:=1+y(x)+y(x)^2+x*(x^2-4)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {1}{2}-\frac {\sqrt {3}\, \tan \left (\frac {\left (-2 \ln \left (x \right )+\ln \left (x -2\right )+\ln \left (x +2\right )+8 c_1 \right ) \sqrt {3}}{16}\right )}{2} \]

✓ Mathematica. Time used: 1.559 (sec). Leaf size: 66

ode=(1+y[x]+y[x]^2)+x*(x^2-4)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} \left (-1+\sqrt {3} \tan \left (\frac {1}{16} \sqrt {3} \left (-\log \left (4-x^2\right )+2 \log (x)+8 c_1\right )\right )\right )\\ y(x)&\to -\sqrt [3]{-1}\\ y(x)&\to (-1)^{2/3} \end{align*}

✓ Sympy. Time used: 0.373 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x**2 - 4)*Derivative(y(x), x) + y(x)**2 + y(x) + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sqrt {3} \tan {\left (C_{1} + \frac {\sqrt {3} \log {\left (x \right )}}{8} - \frac {\sqrt {3} \log {\left (x^{2} - 4 \right )}}{16} \right )}}{2} - \frac {1}{2} \]