Q15RP Determine the inverse Laplace tr... [FREE SOLUTION]

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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: Q15RP (page 416)

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

Short Answer

Expert verified

Therefore, the solution is

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

Step by step solution

Given Information

The given function value in s domain is

$\frac{2 s^{2} + 3 s - 1}{{(s + 1)}^{2} (s + 2)}$

Use partial fractions

Make partial fraction of the given function, as:

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

On comparing the coefficients with $s = 鈭� 1, 鈭� 2, 0$ respectively we get

$\begin{matrix} A = 1 \\ B = 鈭� 2 \\ C = 1 \end{matrix}$

Thus, the partial fractions are obtained as:

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

Take Inverse Laplace transform

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

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

Hence, the required inverse Laplace transform is

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

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91影视

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

Short Answer

Step by step solution

Given Information

Use partial fractions

Take Inverse Laplace transform

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91影视

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

Probability and Statistics

Calculus

Discrete Mathematics

Pure Maths

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

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