83.5.7 problem 2 (b)

Internal problem ID [21942]
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. VI at page 47
Problem number : 2 (b)
Date solved : Thursday, October 02, 2025 at 08:18:46 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Riccati]

\begin{align*} x^{2} y^{\prime }-y x&=x^{2}-y^{2} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.035 (sec). Leaf size: 19

ode:=x^2*diff(y(x),x)-x*y(x) = x^2-y(x)^2; 
ic:=[y(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {x^{3}-x}{x^{2}+1} \]

✓ Mathematica. Time used: 0.294 (sec). Leaf size: 19

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

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

✓ Sympy. Time used: 0.183 (sec). Leaf size: 14

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

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