8.5.27 problem 27

Internal problem ID [755]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 27
Date solved : Saturday, March 29, 2025 at 10:19:52 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

\begin{align*} 3 x y^{2} y^{\prime }&=3 x^{4}+y^{3} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 55

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

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

✓ Mathematica. Time used: 0.21 (sec). Leaf size: 72

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

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

✓ Sympy. Time used: 1.224 (sec). Leaf size: 58

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

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