28.1.23 problem 23

Internal problem ID [4329]
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 : 23
Date solved : Tuesday, March 04, 2025 at 06:24:06 PM
CAS classification : [_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class A`]]

\begin{align*} 3 y \left (x^{2}-1\right )+\left (x^{3}+8 y-3 x \right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 67

ode:=3*y(x)*(x^2-1)+(x^3+8*y(x)-3*x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y \left (x \right ) &= -\frac {x^{3}}{8}+\frac {3 x}{8}-\frac {\sqrt {x^{6}-6 x^{4}+9 x^{2}-16 c_{1}}}{8} \\ y \left (x \right ) &= -\frac {x^{3}}{8}+\frac {3 x}{8}+\frac {\sqrt {x^{6}-6 x^{4}+9 x^{2}-16 c_{1}}}{8} \\ \end{align*}

✓ Mathematica. Time used: 0.177 (sec). Leaf size: 86

ode=(3*y[x]*(x^2-1))+(x^3+8*y[x]-3*x)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{8} \left (-x^3-\sqrt {x^6-6 x^4+9 x^2+64 c_1}+3 x\right ) \\ y(x)\to \frac {1}{8} \left (-x^3+\sqrt {x^6-6 x^4+9 x^2+64 c_1}+3 x\right ) \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.408 (sec). Leaf size: 63

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

\[ \left [ y{\left (x \right )} = - \frac {x \left (x^{2} - 3\right )}{8} + \frac {\sqrt {C_{1} + x^{6} - 6 x^{4} + 9 x^{2}}}{8}, \ y{\left (x \right )} = - \frac {x^{3}}{8} + \frac {3 x}{8} - \frac {\sqrt {C_{1} + x^{6} - 6 x^{4} + 9 x^{2}}}{8}\right ] \]