83.8.5 problem 5

Internal problem ID [21958]
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. IX at page 55
Problem number : 5
Date solved : Thursday, October 02, 2025 at 08:19:58 PM
CAS classification : [_separable]

\begin{align*} x y^{\prime }-y&=y^{3} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 30

ode:=x*diff(y(x),x)-y(x) = y(x)^3; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x}{\sqrt {-x^{2}+c_1}} \\ y &= -\frac {x}{\sqrt {-x^{2}+c_1}} \\ \end{align*}

✓ Mathematica. Time used: 0.375 (sec). Leaf size: 110

ode=x*D[y[x],x]-y[x]==y[x]^3 ; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {i e^{c_1} x}{\sqrt {-1+e^{2 c_1} x^2}}\\ y(x)&\to \frac {i e^{c_1} x}{\sqrt {-1+e^{2 c_1} x^2}}\\ y(x)&\to 0\\ y(x)&\to -i\\ y(x)&\to i\\ y(x)&\to -\frac {i x}{\sqrt {x^2}}\\ y(x)&\to \frac {i x}{\sqrt {x^2}} \end{align*}

✓ Sympy. Time used: 0.533 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) - y(x)**3 - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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