23.2.168 problem 171

Internal problem ID [5523]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 171
Date solved : Tuesday, September 30, 2025 at 12:50:27 PM
CAS classification : [_quadrature]

\begin{align*} \left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}&=b^{2} \end{align*}

✓ Maple. Time used: 0.035 (sec). Leaf size: 44

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

\begin{align*} y &= b \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_1 \\ y &= -b \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 52

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

\begin{align*} y(x)&\to b \arctan \left (\frac {x}{\sqrt {a^2-x^2}}\right )+c_1\\ y(x)&\to -b \arctan \left (\frac {x}{\sqrt {a^2-x^2}}\right )+c_1 \end{align*}

✓ Sympy. Time used: 0.858 (sec). Leaf size: 41

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

\[ \left [ y{\left (x \right )} = C_{1} - b \int \sqrt {- \frac {1}{- a^{2} + x^{2}}}\, dx, \ y{\left (x \right )} = C_{1} + b \int \sqrt {- \frac {1}{- a^{2} + x^{2}}}\, dx\right ] \]