86.1.6 problem 6

Internal problem ID [23068]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 3. Some standard types of differential equations. Exercise 3b at page 43
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:18:39 PM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 5

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

\[ y = x \]

✓ Mathematica. Time used: 0.354 (sec). Leaf size: 6

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

\begin{align*} y(x)&\to x \end{align*}

✓ Sympy. Time used: 0.315 (sec). Leaf size: 3

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

\[ y{\left (x \right )} = x \]