Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Solve the initial value problem in $f'' (t) + 9 f (t) = 0; f (0) = 0, f (\frac{ττ}{2}) = 1$

The solution is. $f (t) = ‐ \sin (3 t)$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of characteristic polynomial

Consider the linear differential operator

$T (f) = f^{(n)} + a_{n - 1} f^{(n - 1)} + . . . + a_{1} f' + a_{0} f .$ .

The characteristic polynomial of T is defined as

$p_{T} (λ) = λ^{n} + a_{n - 1} λ + . . . + a_{1} λ + a_{0}$ .

Step 2: Determination of the solution

The characteristic polynomial of the operator as follows.

$T (f) = f'' + 9$

Then the characteristic polynomial is as follows.

$P_{T} (λ) = λ^{2} + 9$

Solve the characteristic polynomial and find the roots as follows.

$λ^{2} + 9 = 0 λ^{2} = ‐ 9 λ = \sqrt{‐ 9} λ = \pm 3 i$

Therefore, the roots of the characteristic polynomials are $3 i$ and $‐ 3 i$ .

Step 3: Explanation of the solution

Since, the roots of the characteristic equation are different complex numbers.

The exponential functions $e^{3 it}$ and $e^{- 3 it}$ form a basis of the kernel of T .

Hence, they form a basis of the solution space of the homogenous differential equation is $T (f) = 0$ .

Thus, the general solution of the differential equation $f'' (t) + 9 f' (t) = 0$ is $f (t) = c_{1} e^{3 it} + c_{2} e^{- 3 it}$ .

Step 4: Explanation of the solution with initial value problem

Consider the general solution of the $f'' (t) + 9 f' (t) = 0$ with the initial value problem $f (0) = 0, f (\frac{π}{2}) = 1$ is as follows,

$f (t) = c_{1} e^{3 it} + c_{2} e^{- 3 it} f (t) = 0 (e^{3 it}) + 1 (e^{- 3 it}) f (t) = 0 + e^{- 3 it} f (t) = e^{- 3 it}$

Simplify further,

$f (t) = - \sin (3 t)$

Hence the solution is $f (t) = - \sin (3 t)$ .

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Linear Algebra With Applications

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Short Answer

Solve the initial value problem in $f'' (t) + 9 f (t) = 0; f (0) = 0, f (\frac{ττ}{2}) = 1$

Step by Step Solution

Step 1: Definition of characteristic polynomial

Step 2: Determination of the solution

Step 3: Explanation of the solution

Step 4: Explanation of the solution with initial value problem

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Linear Algebra With Applications

Answers without the blur.

Short Answer

Solve the initial value problem in f''(t)+9f(t)=0;f(0)=0,f(ττ2)=1

Step by Step Solution

Step 1: Definition of characteristic polynomial

Step 2: Determination of the solution

Step 3: Explanation of the solution

Step 4: Explanation of the solution with initial value problem

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Solve the initial value problem in $f'' (t) + 9 f (t) = 0; f (0) = 0, f (\frac{ττ}{2}) = 1$