47.2.11 problem 11

Internal problem ID [7427]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 11
Date solved : Sunday, March 30, 2025 at 12:01:03 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 27

ode:=x*diff(y(x),x)-(x^2-y(x)^2)^(1/2)-y(x) = 0; 
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: 0.252 (sec). Leaf size: 18

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

\[ y(x)\to -x \cosh (i \log (x)+c_1) \]

✓ Sympy. Time used: 1.041 (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)
 

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