20.4.19 problem Problem 28

Internal problem ID [3654]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page 79
Problem number : Problem 28
Date solved : Tuesday, March 04, 2025 at 05:01:49 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 29

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

\[ -\arctan \left (\frac {y \left (x \right )}{\sqrt {-y \left (x \right )^{2}+4 x^{2}}}\right )+\ln \left (x \right )-c_{1} = 0 \]

✓ Mathematica. Time used: 0.37 (sec). Leaf size: 18

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

\[ y(x)\to -2 x \cosh (i \log (x)+c_1) \]

✓ Sympy. Time used: 0.969 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = - 2 x \sin {\left (C_{1} - \log {\left (x \right )} \right )} \]