Q37E Find the functionf(t)聽of the fo... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 1: Q37E (page 7)

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

Short Answer

Expert verified

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 $f (t) = 2 e^{3 t} - e^{2 t} .$

Step by step solution

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

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

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.

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

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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91影视

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

Short Answer

Step by step solution

Consider the points and substitute these in the standard equation

Consider the derivative of the polynomial

Solve the above equations.

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91影视

Find the functionf(t)of the form f(t)=ae3t+be2tsuch that f(0)=1andf'(0)=4.

Short Answer

Step by step solution

Consider the points and substitute these in the standard equation

Consider the derivative of the polynomial

Solve the above equations.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Discrete Mathematics

Mechanics Maths

Decision Maths

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

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