78.4.20 problem 21

Internal problem ID [18072]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 8 (Exact Equations). Problems at page 72
Problem number : 21
Date solved : Monday, March 31, 2025 at 05:04:08 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

\begin{align*} \frac {4 y^{2}-2 x^{2}}{4 x y^{2}-x^{3}}+\frac {\left (8 y^{2}-x^{2}\right ) y^{\prime }}{4 y^{3}-x^{2} y}&=0 \end{align*}

✓ Maple. Time used: 0.069 (sec). Leaf size: 71

ode:=(4*y(x)^2-2*x^2)/(4*x*y(x)^2-x^3)+(8*y(x)^2-x^2)/(4*y(x)^3-x^2*y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {\frac {2 c_1 \,x^{3}-2 \sqrt {c_1^{2} x^{6}+16}}{c_1 \,x^{3}}}\, x}{4} \\ y &= \frac {\sqrt {2}\, \sqrt {\frac {c_1 \,x^{3}+\sqrt {c_1^{2} x^{6}+16}}{c_1 \,x^{3}}}\, x}{4} \\ \end{align*}

✓ Mathematica. Time used: 11.817 (sec). Leaf size: 297

ode=(  (4*y[x]^2-2*x^2 )/( 4*x*y[x]^2 - x^3) )+(  (8*y[x]^2-x^2)/(4*y[x]^3-x^2*y[x]) )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\sqrt {x^2-\frac {\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} \\ y(x)\to \frac {\sqrt {x^2-\frac {\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} \\ y(x)\to -\frac {\sqrt {\frac {x^3+\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} \\ y(x)\to \frac {\sqrt {\frac {x^3+\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} \\ y(x)\to -\frac {\sqrt {x^2-\frac {\sqrt {x^6}}{x}}}{2 \sqrt {2}} \\ y(x)\to \frac {\sqrt {x^2-\frac {\sqrt {x^6}}{x}}}{2 \sqrt {2}} \\ y(x)\to -\frac {\sqrt {\frac {\sqrt {x^6}+x^3}{x}}}{2 \sqrt {2}} \\ y(x)\to \frac {\sqrt {\frac {\sqrt {x^6}+x^3}{x}}}{2 \sqrt {2}} \\ \end{align*}

✓ Sympy. Time used: 5.802 (sec). Leaf size: 105

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-2*x**2 + 4*y(x)**2)/(-x**3 + 4*x*y(x)**2) + (-x**2 + 8*y(x)**2)*Derivative(y(x), x)/(-x**2*y(x) + 4*y(x)**3),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {x^{2} - \frac {\sqrt {C_{1} + x^{6}}}{x}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {x^{2} - \frac {\sqrt {C_{1} + x^{6}}}{x}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {x^{2} + \frac {\sqrt {C_{1} + x^{6}}}{x}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {x^{2} + \frac {\sqrt {C_{1} + x^{6}}}{x}}}{4}\right ] \]