23.1.144 problem 145 (b)

Internal problem ID [4751]
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 (b)
Date solved : Tuesday, September 30, 2025 at 08:30:30 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: 43

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: 0.963 (sec). Leaf size: 90

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)
 

\[ y{\left (x \right )} = C_{1} - \begin {cases} \frac {a^{2}}{x \sqrt {\frac {a^{2}}{x^{2}} - 1}} - a \operatorname {acosh}{\left (\frac {a}{x} \right )} - \frac {x}{\sqrt {\frac {a^{2}}{x^{2}} - 1}} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\- \frac {i a^{2}}{x \sqrt {- \frac {a^{2}}{x^{2}} + 1}} + i a \operatorname {asin}{\left (\frac {a}{x} \right )} + \frac {i x}{\sqrt {- \frac {a^{2}}{x^{2}} + 1}} & \text {otherwise} \end {cases} \]