44.5.21 problem 21

Internal problem ID [7083]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.2 Separable equations. Exercises 2.2 at page 53
Problem number : 21
Date solved : Sunday, March 30, 2025 at 11:38:54 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=x \sqrt {1-y^{2}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 12

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

\[ y = \sin \left (\frac {x^{2}}{2}+c_1 \right ) \]

✓ Mathematica. Time used: 0.175 (sec). Leaf size: 34

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

\begin{align*} y(x)\to \sin \left (\frac {x^2}{2}+c_1\right ) \\ y(x)\to -1 \\ y(x)\to 1 \\ y(x)\to \text {Interval}[\{-1,1\}] \\ \end{align*}

✓ Sympy. Time used: 0.286 (sec). Leaf size: 10

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

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