20.4.33 problem Problem 49

Internal problem ID [3668]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page 79
Problem number : Problem 49
Date solved : Tuesday, March 04, 2025 at 05:04:42 PM
CAS classification : [_Bernoulli]

\begin{align*} 2 y^{\prime }+y \cot \left (x \right )&=\frac {8 \cos \left (x \right )^{3}}{y} \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 40

ode:=2*diff(y(x),x)+y(x)*cot(x) = 8/y(x)*cos(x)^3; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y \left (x \right ) &= \csc \left (x \right ) \sqrt {\sin \left (x \right ) \left (-2 \cos \left (x \right )^{4}+c_{1} \right )} \\ y \left (x \right ) &= -\csc \left (x \right ) \sqrt {\sin \left (x \right ) \left (-2 \cos \left (x \right )^{4}+c_{1} \right )} \\ \end{align*}

✓ Mathematica. Time used: 3.926 (sec). Leaf size: 47

ode=2*D[y[x],x]+y[x]*Cot[x]==8/y[x]*Cos[x]^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\sqrt {-2 \cos ^3(x) \cot (x)+c_1 \csc (x)} \\ y(x)\to \sqrt {-2 \cos ^3(x) \cot (x)+c_1 \csc (x)} \\ \end{align*}

✓ Sympy. Time used: 1.141 (sec). Leaf size: 36

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

\[ \left [ y{\left (x \right )} = - \sqrt {\frac {C_{1} - 2 \cos ^{4}{\left (x \right )}}{\sin {\left (x \right )}}}, \ y{\left (x \right )} = \sqrt {\frac {C_{1} - 2 \cos ^{4}{\left (x \right )}}{\sin {\left (x \right )}}}\right ] \]