8.6.27 problem 27

Internal problem ID [797]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Chapter 1 review problems. Page 78
Problem number : 27
Date solved : Saturday, March 29, 2025 at 10:28:45 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

\begin{align*} 3 y+x^{3} y^{4}+3 x y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 56

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

\begin{align*} y &= \frac {1}{\left (\ln \left (x \right )+c_1 \right )^{{1}/{3}} x} \\ y &= -\frac {1+i \sqrt {3}}{2 \left (\ln \left (x \right )+c_1 \right )^{{1}/{3}} x} \\ y &= \frac {i \sqrt {3}-1}{2 \left (\ln \left (x \right )+c_1 \right )^{{1}/{3}} x} \\ \end{align*}

✓ Mathematica. Time used: 0.602 (sec). Leaf size: 70

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

\begin{align*} y(x)\to \frac {1}{\sqrt [3]{x^3 (\log (x)+c_1)}} \\ y(x)\to -\frac {\sqrt [3]{-1}}{\sqrt [3]{x^3 (\log (x)+c_1)}} \\ y(x)\to \frac {(-1)^{2/3}}{\sqrt [3]{x^3 (\log (x)+c_1)}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 2.452 (sec). Leaf size: 68

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

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