76.5.8 problem 8

Internal problem ID [17357]
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 : 8
Date solved : Monday, March 31, 2025 at 03:59:04 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} x y^{\prime }-4 \sqrt {y^{2}-x^{2}}&=y \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 28

ode:=x*diff(y(x),x)-4*(y(x)^2-x^2)^(1/2) = y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {-c_1 \,x^{5}+y+\sqrt {y^{2}-x^{2}}}{x^{5}} = 0 \]

✓ Mathematica. Time used: 0.245 (sec). Leaf size: 16

ode=x*D[y[x],x]-4*Sqrt[ y[x]^2-x^2]==y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -x \cosh (4 \log (x)+c_1) \]

✓ Sympy. Time used: 5.912 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) - 4*sqrt(-x**2 + y(x)**2) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x \cosh {\left (C_{1} - \log {\left (x^{4} \right )} \right )} \]