Q39E Find all solution of the liner D... [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 9: Q39E (page 442)

Find all solution of the liner DE $f''' (t) + 3 f' (t) 3 f' (t) + f (t) = 0$ .

Short Answer

Expert verified

The solution is $f (t) = e^{- t} (c_{1} + c_{2} t + c_{3} t^{2})$ .

Step by step solution

Determine the auxiliary equation of f'''(t)+3f'(t)3f'(t)+f(t)=0 .

Consider the linear differential equation $f''' (t) + 3 f' (t) 3 f' (t) + f (t) = 0$ .

Assume $D^{n} = f^{n} (t)$ , substitute the value $D^{3}$ for $f'' (t)$ , $D^{2}$ for $f'' (t)$ , D for $f' (t)$ and 1 for $f (t)$ in the equation $f''' (t) + 3 f'' (t) + 3 f' (t) + f (t) = 0$ as follows.

$f''' (t) + 3 f'' (t) + 3 f' (t) + f (t) = 0 D^{3} + 3 D^{2} + 3 D + 1 = 0 {(D + 1)}^{3} = 0$

Determine the solution of (D+1)3=0 .

The solution of the differential equation ${(D 卤 a)}^{n} = 0$ isrole="math" localid="1660805138475" $f (t) = e^{T at} \{{鈭�}_{i = 0}^{n - 1} c_{i + 1} t^{i}\}$ .

Using the definition, the solution of the linear differential equation ${(D + 1)}^{3} = 0$ is defined as follows.

$f (t) = e^{- t} \{{鈭�}_{i = 0}^{n - 1} c_{i + 1} t^{i}\} f (t) = e^{- t} \{c_{1} t^{0} + c_{2} t^{1} + c_{3} t^{2}\} f (t) = e^{- t} \{c_{1} + c_{2} t^{1} + c_{3} t^{2}\}$

Hence, the solution of the linear differential equation $f''' (t) + 3 f'' (t) + 3 f' (t) + f (t) = 0$ is

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

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91影视

Find all solution of the liner DE $f''' (t) + 3 f' (t) 3 f' (t) + f (t) = 0$ .

Short Answer

Step by step solution

Determine the auxiliary equation of f'''(t)+3f'(t)3f'(t)+f(t)=0 .

Determine the solution of (D+1)3=0 .

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91影视

Find all solution of the liner DE f'''(t)+3f'(t)3f'(t)+f(t)=0.

Short Answer

Step by step solution

Determine the auxiliary equation of f'''(t)+3f'(t)3f'(t)+f(t)=0 .

Determine the solution of (D+1)3=0 .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Calculus

Geometry

Discrete Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

Find all solution of the liner DE $f''' (t) + 3 f' (t) 3 f' (t) + f (t) = 0$ .