40.2.13 problem 38

Internal problem ID [6591]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 4. Equations of first order and first degree (Variable separable). Supplemetary problems. Page 22
Problem number : 38
Date solved : Sunday, March 30, 2025 at 11:10:58 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 30

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

\[ \frac {-c_1 \,x^{2}+\sqrt {x^{2}+4 y^{2}}+2 y}{x^{2}} = 0 \]

✓ Mathematica. Time used: 0.341 (sec). Leaf size: 18

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

\[ y(x)\to \frac {1}{2} x \sinh (\log (x)+2 c_1) \]

✓ Sympy. Time used: 1.237 (sec). Leaf size: 14

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

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