Q12RP 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: Q12RP (page 416)

Determine the inverse Laplace transform of the given function.
$\frac{2 s - 1}{s^{2} - 4 s + 6}$

Short Answer

Expert verified

The solution is

$L^{鈭� 1} \{\frac{2 (s 鈭� 2) + 3}{{(s 鈭� 2)}^{2} + {(\sqrt{2})}^{2}}\} (t) = 2 e^{2 t} \cos \sqrt{2} t + \frac{3}{\sqrt{2}} e^{2 t} \sin \sqrt{2} t$

Step by step solution

Given Information

The given function value in s domain is $\frac{2 s - 1}{s^{2} - 4 s + 6}$

Use partial fractions

Factorize the denominator of the given function as

$\begin{matrix} s^{2} 鈭� 4 s + 6 = s^{2} 鈭� 4 s + 4 + 2 \\ = {(s - 1)}^{2} + {(\sqrt{2})}^{2} \end{matrix}$

The function becomes. $\frac{2 s - 1}{{(s 鈭� 2)}^{2} + {(\sqrt{2})}^{2}}$ Decompose the function as:

$\begin{matrix} \frac{2 s - 1}{{(s 鈭� 2)}^{2} + {(\sqrt{2})}^{2}} = \frac{2 (s 鈭� 2) + 3}{{(s 鈭� 2)}^{2} + {(\sqrt{2})}^{2}} \\ = \frac{2 (s 鈭� 2)}{{(s 鈭� 2)}^{2} + {(\sqrt{2})}^{2}} + \frac{3}{\sqrt{2}} \frac{\sqrt{2}}{{(s 鈭� 2)}^{2} + {(\sqrt{2})}^{2}} \end{matrix}$

Take inverse Laplace transform using $L^{鈭� 1} \{\frac{s 鈭� a}{{(s 鈭� a)}^{2} + b^{2}}\} (t) = e^{a t} \cos b t$ and $L^{鈭� 1} \{\frac{b}{{(s 鈭� a)}^{2} + b^{2}}\} (t) = e^{a t} \sin b t$ as:

$L^{鈭� 1} \{\frac{2 (s 鈭� 2) + 3}{{(s 鈭� 2)}^{2} + {(\sqrt{2})}^{2}}\} (t) = 2 e^{2 t} \cos \sqrt{2} t + \frac{3}{\sqrt{2}} e^{2 t} \sin \sqrt{2} t$

Therefore,the required inverse Laplace transform is:

$L^{鈭� 1} \{\frac{2 (s 鈭� 2) + 3}{{(s 鈭� 2)}^{2} + {(\sqrt{2})}^{2}}\} (t) = 2 e^{2 t} \cos \sqrt{2} t + \frac{3}{\sqrt{2}} e^{2 t} \sin \sqrt{2} t$

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.
$\frac{2 s - 1}{s^{2} - 4 s + 6}$

Short Answer

Step by step solution

Given Information

Use partial fractions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Discrete Mathematics

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

91影视

Determine the inverse Laplace transform of the given function.2s-1s2-4s+6

Short Answer

Step by step solution

Given Information

Use partial fractions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Discrete Mathematics

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

Determine the inverse Laplace transform of the given function.
$\frac{2 s - 1}{s^{2} - 4 s + 6}$