71.8.37 problem 13 (a)

Internal problem ID [14392]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.4.4, page 115
Problem number : 13 (a)
Date solved : Saturday, February 22, 2025 at 03:44:55 PM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 5

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

\[ y = 1 \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 6

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

\[ y(x)\to 1 \]

✓ Sympy. Time used: 0.292 (sec). Leaf size: 8

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

\[ y{\left (x \right )} = \cos {\left (\frac {x^{2}}{2} \right )} \]