76.5.5 problem 5

Internal problem ID [17346]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.7 (Substitution Methods). Problems at page 108
Problem number : 5
Date solved : Thursday, March 13, 2025 at 09:30:08 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 27

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

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

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

✓ Sympy. Time used: 0.870 (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)
 

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