15.1.7 problem 7

Internal problem ID [2847]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 5, page 21
Problem number : 7
Date solved : Sunday, March 30, 2025 at 12:34:03 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 34

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

\[ c_1 +x \sqrt {-x^{2}+1}+\arcsin \left (x \right )+y \sqrt {1-y^{2}}+\arcsin \left (y\right ) = 0 \]

✓ Mathematica. Time used: 0.525 (sec). Leaf size: 55

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

\[ y(x)\to \text {InverseFunction}\left [\frac {1}{2} \left (\text {$\#$1} \sqrt {1-\text {$\#$1}^2}+\arcsin (\text {$\#$1})\right )\&\right ]\left [-\frac {\arcsin (x)}{2}-\frac {1}{2} \sqrt {1-x^2} x+c_1\right ] \]

✓ Sympy. Time used: 1.832 (sec). Leaf size: 39

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

\[ \frac {x \sqrt {1 - x^{2}}}{2} + \frac {\sqrt {1 - y^{2}{\left (x \right )}} y{\left (x \right )}}{2} + \frac {\operatorname {asin}{\left (x \right )}}{2} + \frac {\operatorname {asin}{\left (y{\left (x \right )} \right )}}{2} = C_{1} \]