20.4.36 problem Problem 52

Internal problem ID [3671]
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 52
Date solved : Tuesday, March 04, 2025 at 05:05:31 PM
CAS classification : [_Bernoulli]

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

With initial conditions

\begin{align*} y \left (\frac {\pi }{2}\right )&=1 \end{align*}

✓ Maple. Time used: 3.622 (sec). Leaf size: 34

ode:=diff(y(x),x)+y(x)*cot(x) = y(x)^3*sin(x)^3; 
ic:=y(1/2*Pi) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.946 (sec). Leaf size: 20

ode=D[y[x],x]+y[x]*Cot[x]==y[x]^3*Sin[x]^3; 
ic={y[Pi/2]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{\sqrt {\sin ^2(x) (2 \cos (x)+1)}} \]

✓ Sympy. Time used: 0.743 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)**3*sin(x)**3 + y(x)/tan(x) + Derivative(y(x), x),0) 
ics = {y(pi/2): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sqrt {\frac {1}{2 \cos {\left (x \right )} + 1}}}{\sin {\left (x \right )}} \]