63.5.2 problem 1(b)

Internal problem ID [13005]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.4.1. Integrating factors. Exercises page 41
Problem number : 1(b)
Date solved : Monday, March 31, 2025 at 07:30:28 AM
CAS classification : [_separable]

\begin{align*} \cos \left (t \right ) x^{\prime }-2 x \sin \left (x\right )&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 24

ode:=cos(t)*diff(x(t),t)-2*x(t)*sin(x(t)) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ \ln \left (\sec \left (t \right )+\tan \left (t \right )\right )-\frac {\int _{}^{x}\frac {\csc \left (\textit {\_a} \right )}{\textit {\_a}}d \textit {\_a}}{2}+c_1 = 0 \]

✓ Mathematica. Time used: 0.282 (sec). Leaf size: 40

ode=Cos[t]*D[x[t],t]-2*x[t]*Sin[x[t]]==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\csc (K[1])}{K[1]}dK[1]\&\right ]\left [4 \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )+c_1\right ] \\ x(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.539 (sec). Leaf size: 24

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

\[ \int \limits ^{x{\left (t \right )}} \frac {1}{y \sin {\left (y \right )}}\, dy = C_{1} - \log {\left (\sin {\left (t \right )} - 1 \right )} + \log {\left (\sin {\left (t \right )} + 1 \right )} \]