Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find the function $f (t)$ of the form $f (t) = a e^{3 t} + b e^{2 t}$ such that $f (0) = 1$ and $f' (0) = 4$ .

The function of the form $f (t) = a e^{3 t} + b e^{2 t}$ such that $f (0) = 1$ and $f' (0) = 4$ is $f (t) = 2 e^{3 t} - e^{2 t} .$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Consider the points and substitute these in the standard equation

Consider the function $f (t) = a e^{3 t} + b e^{2 t}$ . Put $f (0) = 1$ in $f (t) = a e^{3 t} + b e^{2 t}$

$f (0) = a e^{3 (0)} + b e^{2 (0)} 1 = a + b$

Step 2: Consider the derivative of the polynomial

Take the derivative of the polynomial $f (t) = a e^{3 t} + b e^{2 t}$ and put $f' (0) = 4$ into its derivative.

$f^{'} = (t) = 3 a e^{3 t} + 2 b e^{2 t} f^{'} (0) = 3 a e^{3 \times 0} + 2 b e^{2 \times 0} 4 = 3 a + 2 b$

Step 3: Solve the above equations.

Arrange the equations obtained in above steps into the matrix form.

$(\begin{matrix} 1 & 1 \\ 3 & 2 \end{matrix}) (\begin{matrix} a \\ b \end{matrix}) = (\begin{matrix} 1 \\ 4 \end{matrix})$

Upon solving the values of $a, b$ are obtained as $a = 2, b = - 1$

Substitute these values in the polynomial equation.

role="math" localid="1659340662448" $f (t) = a e^{3 t} + b e^{2 t} f (t) = 2 e^{3 t} - 1 e^{2 t}$

The function $f (t)$ of the form $f (t) = a e^{3 t} + b e^{2 t}$ such that $f (0) = 1$ and $f' (0) = 4$ is $f (t) = 2 e^{3 t} - e^{2 t}$ .

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Linear Algebra With Applications

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Short Answer

Find the function $f (t)$ of the form $f (t) = a e^{3 t} + b e^{2 t}$ such that $f (0) = 1$ and $f' (0) = 4$ .

Step by Step Solution

Step 1: Consider the points and substitute these in the standard equation

Step 2: Consider the derivative of the polynomial

Step 3: Solve the above equations.

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Linear Algebra With Applications

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Short Answer

Find the functionf(t) of the form f(t)=ae3t+be2t such that f(0)=1and f'(0)=4.

Step by Step Solution

Step 1: Consider the points and substitute these in the standard equation

Step 2: Consider the derivative of the polynomial

Step 3: Solve the above equations.

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Find the function $f (t)$ of the form $f (t) = a e^{3 t} + b e^{2 t}$ such that $f (0) = 1$ and $f' (0) = 4$ .