56.12.9 problem Ex 10

Internal problem ID [14136]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 19. Summary. Page 29
Problem number : Ex 10
Date solved : Thursday, October 02, 2025 at 09:15:50 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 27

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

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

✓ Mathematica. Time used: 0.161 (sec). Leaf size: 18

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

\begin{align*} y(x)&\to -x \cosh (i \log (x)+c_1) \end{align*}

✓ Sympy. Time used: 0.552 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = - x \sin {\left (C_{1} - \log {\left (x \right )} \right )} \]