Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

The temperature of a hot cup of coffee can be modelled by the DE
$T' (t) = - k (T (t) - A)$

(a) What is the significance of the constants K and A?
(b) Solve the DE for T (t) in terms of K, A and the initial temperature $T_{0}$

(a) If the constant A is positive then the temperature of the hot cup of coffee modelled by the DE is positive and A is negative then the temperature of the hot cup of coffee modelled by the DE is negative.

(b)The solution is $A = T_{0} - C e^{- k t}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1 Explanation for Linear Differential operators and linear differential equations

A transformation T from $C^{\infty} to C^{\infty}$ to of the form role="math" localid="1660801324664" $T (f) = f^{(n)} + a_{n - 1} f^{(n - 1)} + . . . + a_{1} f' + a_{0} f$ is called an nth-order linear differential operator.Here $f^{(k)}$ denote the k-th derivative of function f and the coefficients $a_{k}$ are complex scalars.

If T is an nth-order linear differential operator and g is a smooth function, then the equation becomes

$T (f) = g$

(or)

$f^{(n)} + a_{n - 1} f^{(n - 1)} + . . . + a_{1} f' + a_{0} f = g$

Is called an nth-order linear differential equation (DE).The DE is called homogeneous if g = 0 and inhomogeneous otherwise.

(a) Step 2: Explanation of the significance for the constants K and A

Consider the temperature of a hot cup of coffee can be modelled by the DE

$T' (t) = k (T (t) - A)$

Here k and A be the constants in the differential equation.

The constant is called the continuous growth rate if it is positive and the continuous decay rate if it is negative.

If the constant A is positive then the temperature of the hot cup of coffee modelled by the DE is positive and A is negative then the temperature of the hot cup of coffee modelled by the DE is negative.

Since both the constants k and A be the significant terms in the temperature of the hot cup of coffee modelled by the DE

(b) Step 3 : Solution for the DE for T(t) in terms k, A and the initial temperature T0

Consider the temperature of a hot cup of coffee can be modelled by the DE

$T' (t) = k (T (t) - A)$

Here k and A be the constants in the differential equation.

$T' (t) = k (T (t) - A) T' (t) = - k (T (t)) + k A T' (t) = (A - t) k ∵ T' (t) = \frac{d T}{d t} \frac{d T}{d t} = (A - T) k \frac{d T}{d t} = - k (T - A)$

Integrating on both sides we get,

$\frac{d T}{d t} = - k (T - A) \frac{d T}{(T - A)} = - k d t \int \frac{d T}{(T - A)} = \int - k d t \ln |T - A| = - k t + C T - A = e^{- k t} e^{c} T - A = C e^{- k t} T (t) = A + C e^{- k t} ∵ A = T - C e^{- k t}$

Here initial temperature becomes $T_{0}$ and the solution is $A = T_{0} - C e^{- k t}$ .

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Linear Algebra With Applications

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Short Answer

The temperature of a hot cup of coffee can be modelled by the DE
$T' (t) = - k (T (t) - A)$

(a) What is the significance of the constants K and A?
(b) Solve the DE for T (t) in terms of K, A and the initial temperature $T_{0}$

Step by Step Solution

Step 1 Explanation for Linear Differential operators and linear differential equations

(a) Step 2: Explanation of the significance for the constants K and A

(b) Step 3 : Solution for the DE for T(t) in terms k, A and the initial temperature T0

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Linear Algebra With Applications

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Short Answer

The temperature of a hot cup of coffee can be modelled by the DET'(t)=-k(T(t)-A) (a) What is the significance of the constants K and A? (b) Solve the DE for T (t) in terms of K, A and the initial temperature T0

Step by Step Solution

Step 1 Explanation for Linear Differential operators and linear differential equations

(a) Step 2: Explanation of the significance for the constants K and A

(b) Step 3 : Solution for the DE for T(t) in terms k, A and the initial temperature T0

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Pure Maths

The temperature of a hot cup of coffee can be modelled by the DE
$T' (t) = - k (T (t) - A)$

(a) What is the significance of the constants K and A?
(b) Solve the DE for T (t) in terms of K, A and the initial temperature $T_{0}$