23.1.412 problem 399

Internal problem ID [5019]
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 : 399
Date solved : Tuesday, September 30, 2025 at 11:23:28 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 135

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

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

✓ Mathematica. Time used: 12.836 (sec). Leaf size: 101

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

\begin{align*} y(x)&\to -b \sqrt {\sec ^2\left (\frac {b \left (\arctan \left (\frac {\sqrt {x^2-a^2}}{a}\right )+a c_1\right )}{a}\right )}\\ y(x)&\to b \sqrt {\sec ^2\left (\frac {b \left (\arctan \left (\frac {\sqrt {x^2-a^2}}{a}\right )+a c_1\right )}{a}\right )}\\ y(x)&\to 0\\ y(x)&\to -b\\ y(x)&\to b \end{align*}

✓ Sympy. Time used: 1.730 (sec). Leaf size: 34

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

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