8.3.5 problem 5

Internal problem ID [681]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.4. Separable equations. Page 43
Problem number : 5
Date solved : Saturday, March 29, 2025 at 10:12:54 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 12

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

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

✓ Mathematica. Time used: 0.181 (sec). Leaf size: 32

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

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

✓ Sympy. Time used: 0.249 (sec). Leaf size: 10

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

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