67.5.11 problem 6.5 (c)

Internal problem ID [16409]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 6. Simplifying through simplifiction. Additional exercises. page 114
Problem number : 6.5 (c)
Date solved : Thursday, October 02, 2025 at 01:28:59 PM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 17

ode:=diff(y(x),x)+3*cot(x)*y(x) = 6*cos(x)*y(x)^(2/3); 
dsolve(ode,y(x), singsol=all);
 

\[ -\sin \left (x \right )-c_1 \csc \left (x \right )+y^{{1}/{3}} = 0 \]

✓ Mathematica. Time used: 0.18 (sec). Leaf size: 34

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

\begin{align*} y(x)&\to \frac {1}{27} \csc ^3(x) \left (\int _1^x3 \sin (2 K[1])dK[1]+3 c_1\right ){}^3 \end{align*}

✓ Sympy. Time used: 0.279 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = \frac {C_{1}^{3}}{\sin ^{3}{\left (x \right )}} + \frac {3 C_{1}^{2}}{\sin {\left (x \right )}} + 3 C_{1} \sin {\left (x \right )} + \sin ^{3}{\left (x \right )} \]