89.1.16 problem 16

Internal problem ID [24251]
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. Exercises at page 21
Problem number : 16
Date solved : Thursday, October 02, 2025 at 10:02:00 PM
CAS classification : [_separable]

\begin{align*} x^{\prime }&=\sin \left (x\right )^{2} \cos \left (t \right )^{3} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 21

ode:=diff(x(t),t) = sin(x(t))^2*cos(t)^3; 
dsolve(ode,x(t), singsol=all);
 

\[ x = \pi -\operatorname {arccot}\left (c_1 +\frac {\sin \left (3 t \right )}{12}+\frac {3 \sin \left (t \right )}{4}\right ) \]

✓ Mathematica. Time used: 0.342 (sec). Leaf size: 56

ode=D[x[t],t]==Sin[ x[t] ]^2 *Cos[t]^3; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \cot ^{-1}\left (\frac {1}{48} (-36 \sin (t)-4 \sin (3 t)-3 c_1)\right )\\ x(t)&\to -\cot ^{-1}\left (\frac {1}{48} (36 \sin (t)+4 \sin (3 t)+3 c_1)\right )\\ x(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.238 (sec). Leaf size: 19

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

\[ x{\left (t \right )} = - \operatorname {atan}{\left (\frac {3}{C_{1} - \sin ^{3}{\left (t \right )} + 3 \sin {\left (t \right )}} \right )} \]