60.1.87 problem 89

Internal problem ID [10101]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 89
Date solved : Wednesday, March 05, 2025 at 08:30:57 AM
CAS classification : [_quadrature]

\begin{align*} x y^{\prime }-\sqrt {a^{2}-x^{2}}&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 51

ode:=x*diff(y(x),x)-(a^2-x^2)^(1/2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\sqrt {a^{2}-x^{2}}\, \operatorname {csgn}\left (a \right )+a \right )}{x}\right )-a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )+\sqrt {a^{2}-x^{2}}+c_{1} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 42

ode=x*D[y[x],x] - Sqrt[a^2 - x^2]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -a \text {arctanh}\left (\frac {\sqrt {a^2-x^2}}{a}\right )+\sqrt {a^2-x^2}+c_1 \]

✓ Sympy. Time used: 1.324 (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)
 

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