Q14RP Determine the inverse Laplace tr... [FREE SOLUTION]

91影视

Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 7: Q14RP (page 416)

Determine the inverse Laplace transform of the given function.
$\frac{s^{2} + 16 s + 9}{(s + 1) (s + 3) (s - 2)}$

Short Answer

Expert verified

Therefore, the solution is

$L^{鈭� 1} {\frac{s^{2} + 16 s + 9}{(s + 1) (s + 3) (s 鈭� 2)}} (t) = e^{鈭� t} 鈭� 3 e^{鈭� 3 t} + 3 e^{2 t}$

Step by step solution

Given Information

The given function value in s domain is $\frac{s^{2} + 16 s + 9}{(s + 1) (s + 3) (s - 2)}$

Use partial fractions

Make partial fraction of the given function, as:

$\begin{matrix} \frac{s^{2} + 16 s + 9}{(s + 1) (s + 3) (s 鈭� 2)} = \frac{A}{s + 1} + \frac{B}{s + 3} + \frac{C}{s 鈭� 2} \\ = \frac{A (s + 3) (s 鈭� 2) + B (s + 1) (s 鈭� 2) + C (s + 1) (s + 3)}{(s + 1) (s + 3) (s 鈭� 2)} \end{matrix}$

On comparing the coefficients with $s = 鈭� 1, 鈭� 3, 2$ respectively we get

$\begin{array}{l} A = 1 \\ B = 鈭� 3 \\ C = 3 \end{array}$

Thus, the partial fractions are obtained as:

$\frac{s^{2} + 16 s + 9}{(s + 1) (s + 3) (s 鈭� 2)} = \frac{1}{s + 1} 鈭� \frac{3}{s + 3} + \frac{3}{s 鈭� 2}$

Take Inverse Laplace transform

Take inverse Laplace transform using $L^{鈭� 1} {\frac{1}{s 鈭� a}} (t) = e^{a t}$ as:

$\begin{matrix} L^{鈭� 1} {\frac{1}{s + 1} 鈭� \frac{3}{s + 3} + \frac{3}{s 鈭� 2}} = L^{鈭� 1} {\frac{1}{s + 1}} (t) 鈭� 3 L^{鈭� 1} {\frac{1}{s + 3}} (t) + 3 L^{鈭� 1} {\frac{1}{s 鈭� 2}} (t) \\ = e^{鈭� t} 鈭� 3 e^{鈭� 3 t} + 3 e^{2 t} \end{matrix}$

Hence, the required inverse Laplace transform is

$L^{鈭� 1} {\frac{s^{2} + 16 s + 9}{(s + 1) (s + 3) (s 鈭� 2)}} (t) = e^{鈭� t} 鈭� 3 e^{鈭� 3 t} + 3 e^{2 t}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Determine the inverse Laplace transform of the given function.
$\frac{s^{2} + 16 s + 9}{(s + 1) (s + 3) (s - 2)}$

Short Answer

Step by step solution

Given Information

Use partial fractions

Take Inverse Laplace transform

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Statistics

Probability and Statistics

Pure Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

91影视

Determine the inverse Laplace transform of the given function.s2+16s+9(s+1)(s+3)(s-2)

Short Answer

Step by step solution

Given Information

Use partial fractions

Take Inverse Laplace transform

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Statistics

Probability and Statistics

Pure Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Determine the inverse Laplace transform of the given function.
$\frac{s^{2} + 16 s + 9}{(s + 1) (s + 3) (s - 2)}$