12.3.24 problem 25

Internal problem ID [1601]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. separable equations. Section 2.2 Page 52
Problem number : 25
Date solved : Saturday, March 29, 2025 at 11:06:31 PM
CAS classification : [_separable]

\begin{align*} y^{\prime } \sqrt {-x^{2}+1}+\sqrt {1-y^{2}}&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 11

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

\[ y = -\sin \left (\arcsin \left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.185 (sec). Leaf size: 33

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

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

✓ Sympy. Time used: 0.280 (sec). Leaf size: 8

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

\[ y{\left (x \right )} = \sin {\left (C_{1} - \operatorname {asin}{\left (x \right )} \right )} \]