89.6.19 problem 19

Internal problem ID [24402]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Miscellaneous Exercises at page 45
Problem number : 19
Date solved : Thursday, October 02, 2025 at 10:25:08 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.087 (sec). Leaf size: 69

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

\[ x = \arctan \left (\frac {c_1^{2} {\mathrm e}^{t +\frac {\sin \left (2 t \right )}{2}}-1}{c_1^{2} {\mathrm e}^{t +\frac {\sin \left (2 t \right )}{2}}+1}, \frac {2 c_1 \,{\mathrm e}^{\frac {t}{2}+\frac {\sin \left (2 t \right )}{4}}}{c_1^{2} {\mathrm e}^{t +\frac {\sin \left (2 t \right )}{2}}+1}\right ) \]

✓ Mathematica. Time used: 0.776 (sec). Leaf size: 41

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

\begin{align*} x(t)&\to 2 \arctan \left (\tanh \left (\frac {1}{8} (2 t+\sin (2 t)+c_1)\right )\right )\\ x(t)&\to -\frac {\pi }{2}\\ x(t)&\to \frac {\pi }{2} \end{align*}

✗ Sympy

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

Timed Out