23.1.143 problem 145 (a)

Internal problem ID [4750]
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 : 145 (a)
Date solved : Tuesday, September 30, 2025 at 08:30:29 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 51

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

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 42

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

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

✓ Sympy. Time used: 0.898 (sec). Leaf size: 19

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

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