8.5.12 problem 12

Internal problem ID [740]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 12
Date solved : Saturday, March 29, 2025 at 10:17:32 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 30

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

\[ \frac {\ln \left (x \right ) x -c_1 x -\sqrt {4 x^{2}+y^{2}}}{x} = 0 \]

✓ Mathematica. Time used: 0.259 (sec). Leaf size: 54

ode=x*y[x]*D[y[x],x] == y[x]^2+x*(4*x^2+y[x]^2)^(1/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -x \sqrt {\log ^2(x)+2 c_1 \log (x)-4+c_1{}^2} \\ y(x)\to x \sqrt {\log ^2(x)+2 c_1 \log (x)-4+c_1{}^2} \\ \end{align*}

✗ Sympy

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

TypeError : cannot determine truth value of Relational