63.1.42 problem 61

Internal problem ID [15482]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 61
Date solved : Thursday, October 02, 2025 at 10:18:23 AM
CAS classification : [_linear]

\begin{align*} s^{\prime }+s \cos \left (t \right )&=\frac {\sin \left (2 t \right )}{2} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 15

ode:=diff(s(t),t)+s(t)*cos(t) = 1/2*sin(2*t); 
dsolve(ode,s(t), singsol=all);
 

\[ s = \sin \left (t \right )-1+{\mathrm e}^{-\sin \left (t \right )} c_1 \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 54

ode=D[s[t],t]+s[t]*Cos[t]==1/2*Sin[2*t]; 
ic={}; 
DSolve[{ode,ic},s[t],t,IncludeSingularSolutions->True]
 

\begin{align*} s(t)&\to \exp \left (\int _1^t-\cos (K[1])dK[1]\right ) \left (\int _1^t\exp \left (-\int _1^{K[2]}-\cos (K[1])dK[1]\right ) \cos (K[2]) \sin (K[2])dK[2]+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.276 (sec). Leaf size: 14

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

\[ s{\left (t \right )} = C_{1} e^{- \sin {\left (t \right )}} + \sin {\left (t \right )} - 1 \]