Answered: dT equation dt k(T A).

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Determine the general soloution to the given differntial equation

Expert Solution

Step 1 given information

we have been given the differential equation $\frac{d T}{d t} = k (T - A)$

this is the first-order linear differential equation

Step 2 calculations

using the variable separable method $\int y (x) d y = \int x d x$ we get

$\frac{d T}{T - A} = k d t$

integrating both sides we get

$\begin{array}{rcl} \int \frac{d T}{T - A} & = & \int k d t \\ \log (T - A) & = & k t + \log c \\ \log (T - A) - \log c & = & k t \\ \log (\frac{T - A}{c}) & = & k t \\ \frac{T - A}{c} & = & e^{k t} (a s \log_{e} y = x \Rightarrow y = e^{x}) \\ T - A & = & c e^{k t} \\ T & = & c . e^{k t} + A \end{array}$

where 'c' is the constant

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