65.4.9 problem 9.4

Internal problem ID [13668]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 9, First order linear equations and the integrating factor. Exercises page 86
Problem number : 9.4
Date solved : Monday, March 31, 2025 at 08:07:14 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} T^{\prime }&=-k \left (T-\mu -a \cos \left (\omega \left (t -\phi \right )\right )\right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 58

ode:=diff(T(t),t) = -k*(T(t)-mu-a*cos(omega*(t-phi))); 
dsolve(ode,T(t), singsol=all);
 

\[ T = \frac {\cos \left (\omega \left (-t +\phi \right )\right ) a \,k^{2}-\sin \left (\omega \left (-t +\phi \right )\right ) a k \omega +\left (k^{2}+\omega ^{2}\right ) \left ({\mathrm e}^{-k t} c_1 +\mu \right )}{k^{2}+\omega ^{2}} \]

✓ Mathematica. Time used: 0.704 (sec). Leaf size: 74

ode=D[ T[t],t]==-k*(T[t]- (mu+a*Cos[ omega*(t-phi)])); 
ic={}; 
DSolve[{ode,ic},T[t],t,IncludeSingularSolutions->True]
 

\[ T(t)\to \frac {a k e^{-i \omega (\phi +t)} \left ((k+i \omega ) e^{2 i \omega \phi }+(k-i \omega ) e^{2 i \omega t}\right )}{2 \left (k^2+\omega ^2\right )}+c_1 e^{-k t}+\mu \]

✓ Sympy. Time used: 0.285 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
a = symbols("a") 
k = symbols("k") 
mu = symbols("mu") 
omega = symbols("omega") 
phi = symbols("phi") 
T = Function("T") 
ode = Eq(k*(-a*cos(omega*(-phi + t)) - mu + T(t)) + Derivative(T(t), t),0) 
ics = {} 
dsolve(ode,func=T(t),ics=ics)
 

\[ T{\left (t \right )} = C_{1} e^{- k t} + \frac {a k^{2} \cos {\left (\omega \left (\phi - t\right ) \right )}}{k^{2} + \omega ^{2}} - \frac {a k \omega \sin {\left (\omega \left (\phi - t\right ) \right )}}{k^{2} + \omega ^{2}} + \mu \]