44.6.30 problem 30

Internal problem ID [7174]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.3 Linear equations. Exercises 2.3 at page 63
Problem number : 30
Date solved : Wednesday, March 05, 2025 at 04:19:58 AM
CAS classification : [_quadrature]

\begin{align*} T^{\prime }&=k \left (T-T_{m} \right ) \end{align*}

With initial conditions

\begin{align*} T \left (0\right )&=T_{0} \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 16

ode:=diff(T(t),t) = k*(T(t)-T__m); 
ic:=T(0) = T__0; 
dsolve([ode,ic],T(t), singsol=all);
 

\[ T = T_{m} +{\mathrm e}^{t k} \left (T_{0} -T_{m} \right ) \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 18

ode=D[T[t],t]==k*(T[t]-Tm); 
ic={T[0]==T0}; 
DSolve[{ode,ic},T[t],t,IncludeSingularSolutions->True]
 

\[ T(t)\to e^{k t} (\text {T0}-\text {Tm})+\text {Tm} \]

✓ Sympy. Time used: 0.164 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
T__m = symbols("T__m") 
k = symbols("k") 
T = Function("T") 
ode = Eq(-k*(-T__m + T(t)) + Derivative(T(t), t),0) 
ics = {T(0): T__0} 
dsolve(ode,func=T(t),ics=ics)
 

\[ T{\left (t \right )} = T^{m} + \left (T^{0} - T^{m}\right ) e^{k t} \]