89.1.22 problem 22

Internal problem ID [24257]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 21
Problem number : 22
Date solved : Thursday, October 02, 2025 at 10:02:11 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 49

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

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 28

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

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

✓ Sympy. Time used: 0.727 (sec). Leaf size: 31

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

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