38.6.30 problem 30

Internal problem ID [8460]
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 : Tuesday, September 30, 2025 at 05:37:27 PM
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: 0.022 (sec). Leaf size: 16

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

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

✓ Mathematica. Time used: 0.02 (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]
 

\begin{align*} T(t)&\to e^{k t} (\text {T0}-\text {Tm})+\text {Tm} \end{align*}

✓ Sympy. Time used: 0.073 (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)
 

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