28.1.76 problem 79

Internal problem ID [4382]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 79
Date solved : Sunday, March 30, 2025 at 03:11:14 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} 6 y^{2}-x \left (2 x^{3}+y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.330 (sec). Leaf size: 64

ode:=6*y(x)^2-x*(2*x^3+y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x^{3} \left (x^{3}-\sqrt {x \left (x^{3}+8 c_1 \right )}\, x +4 c_1 \right )}{2 c_1} \\ y &= \frac {x^{3} \left (x^{3}+\sqrt {x \left (x^{3}+8 c_1 \right )}\, x +4 c_1 \right )}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 1.307 (sec). Leaf size: 123

ode=6*y[x]^2-(x*(2*x^3+y[x]))*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to 2 x^3 \left (-1+\frac {2}{1-\frac {4 x^{3/2}}{\sqrt {16 x^3+c_1}}}\right ) \\ y(x)\to 2 x^3 \left (-1+\frac {2}{1+\frac {4 x^{3/2}}{\sqrt {16 x^3+c_1}}}\right ) \\ y(x)\to 0 \\ y(x)\to 2 x^3 \\ y(x)\to \frac {2 \left (\left (x^3\right )^{3/2}-x^{9/2}\right )}{x^{3/2}+\sqrt {x^3}} \\ \end{align*}

✓ Sympy. Time used: 1.411 (sec). Leaf size: 78

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(2*x**3 + y(x))*Derivative(y(x), x) + 6*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {3 C_{1} \sqrt {x^{9} \left (9 C_{1}^{2} x^{3} + 8\right )}}{2} + \frac {x^{3} \left (9 C_{1}^{2} x^{3} + 4\right )}{2}, \ y{\left (x \right )} = \frac {3 C_{1} \sqrt {x^{9} \left (9 C_{1}^{2} x^{3} + 8\right )}}{2} + \frac {x^{3} \left (9 C_{1}^{2} x^{3} + 4\right )}{2}\right ] \]