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 : Tuesday, February 04, 2025 at 12:44:11 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*}

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

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

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