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Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 375
Fundamentals Of Differential Equations And Boundary Value Problems

Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

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Short Answer

In Problems 1-10, determine the inverse Laplace transform of the given function.

s−12s2+s+6

The inverse Laplace transform for the given function is

L−1s−12s2+s+6=12e−14tcos474t−54794e−14tsin474t

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Determining the inverse laplace transform

  • For a given transfer function H, the Inverse Laplace Transform takes the output Y(s) and determines what X(s) it is in terms of (s).
  • Consider a function F(s), if there is a function f(t) that is continuous on [0,∞)and satisfies L{f}=Fthen we say that f(t) is the inverse Laplace transform of F(s) and employ the notation
  • f=L−1{F}
  • L−1n!(s−a)n+1=eattn,n=1,2,…

Step 2: Find inverse laplace transform for the given function

The given function is s−12s2+s+6

Simplify s−12s2+s+6 as:

s−12s2+s+6=s−12s2+12s+3=12s−1s2+12s+3=12s−1s2+12s+116+3−116=12s−1s+142+47162

Further simplify the equation as follows:

s−12s2+s+6=12s+14−54s+142+4742=12s+14−54s+142+4742=12s+14s+142+4742−54794474s+142+4742

Find the inverse Laplace transform of

s−12s2+s+6=12s+14s+142+4742−54794474s+142+4742using L−1b(s−a)2+(b)2=eatsinbt and L−1s−a(s−a)2+(b)2=eatcosbt as:

L−1s−12s2+s+6=L−112s+14s+142+4742−54794474s+142+4742=L−112s+14s+142+4742−L−154794474s+142+4742=12L−1s+14s+142+4742−54794L−1474s+142+4742=12e−14tcos474t−54794e−14tsin474t

Therefore, the inverse Laplace transform for the given function is

L−1s−12s2+s+6=12e−14tcos474t−54794e−14tsin474t

Most popular questions for Math Textbooks

In Problems 3-10, determine the Laplace transform of the given function.

(t+3)2-(et+3)2

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

cosnt sinmt,m≠n

In Problems 25 - 32, solve the given initial value problem using the method of Laplace transforms.

y''+4y=g(t); y(0)=1, y'(0)=3,where g(t)={sint, 0⩽t⩽2π,0, 2π<t

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

2t2e-t-t+cos4t.

In Problems 1-14, solve the given initial value problem using the method of Laplace transforms

y''+4y=4t2-4t+10; y0=0, y'0=3
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