23.1.729 problem 726

Internal problem ID [5336]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 726
Date solved : Tuesday, September 30, 2025 at 12:30:45 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 75

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

\[ \frac {x \sqrt {a^{2}-x^{2}}}{2}+\frac {a^{2} \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )}{2}-\frac {y \sqrt {b^{2}-y^{2}}}{2}-\frac {b^{2} \arctan \left (\frac {y}{\sqrt {b^{2}-y^{2}}}\right )}{2}+c_1 = 0 \]

✓ Mathematica. Time used: 0.797 (sec). Leaf size: 94

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {1}{2} \left (b^2 \arctan \left (\frac {\text {$\#$1}}{\sqrt {b^2-\text {$\#$1}^2}}\right )+\text {$\#$1} \sqrt {b^2-\text {$\#$1}^2}\right )\&\right ]\left [\frac {1}{2} \left (a^2 \arctan \left (\frac {x}{\sqrt {a^2-x^2}}\right )+x \sqrt {a^2-x^2}\right )+c_1\right ] \end{align*}

✗ Sympy

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

Timed Out