Find inverse Laplace transform ${L}^{-1}\left\{\frac{s-5}{{s}^{2}+5s-24}\right\}$
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pedzenekO
2020-12-27
Answered

Find inverse Laplace transform ${L}^{-1}\left\{\frac{s-5}{{s}^{2}+5s-24}\right\}$
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izboknil3

Answered 2020-12-28
Author has **99** answers

Step 1

The following formulae are used in solving the given problem.

${L}^{-1}f(x)+g(x)={L}^{-1}(f(x))+{L}^{-1}(g(x))$

${L}^{-1}(a\cdot f(x))=a{L}^{-1}(f(x))$

${L}^{-1}\left(\frac{1}{s+a}\right)={e}^{-at}$

Step 2

Evaluate the inverse Laplace transform of the given function as follows.

${L}^{-1}\left\{\frac{s-5}{{s}^{2}+5s-24}\right\}={L}^{-1}\{-\frac{2}{11}(s-3)+\frac{13}{11}(s+8)\}(\text{by partial fractions})$

${L}^{-1}\{-\frac{2}{11(s-3)}\}+{L}^{-1}\left\{\frac{13}{11(s+8)}\right\}$

$-\frac{2}{11}{L}^{-1}\{1(s-3)\}+\frac{13}{11}{L}^{-1}\left\{\frac{1}{s+8}\right\}$

$-\frac{2}{11}{e}^{3t}+\frac{13}{11}{e}^{-8t}$

Therefore, the inverse Laplace transform of the given function is$-\frac{2}{11}{e}^{3t}+\frac{13}{11}{e}^{-8t}$

The following formulae are used in solving the given problem.

Step 2

Evaluate the inverse Laplace transform of the given function as follows.

Therefore, the inverse Laplace transform of the given function is

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$A=\left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ -{a}_{0}& -{a}_{1}& -{a}_{2}& \cdots & -{a}_{n-1}\end{array}\right]$

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${y}^{(n)}+{a}_{n-1}{y}^{(n-1)}+\cdots +{a}_{1}\dot{y}+{a}_{0}y={b}_{n-1}{u}^{(n-1)}+\cdots +{b}_{1}\dot{u}+{b}_{0}u+g(y(t),u(t))$

I want to express the above differential equation into a system of linear differential equations of the form

$\dot{x}=Ax+Bu+{B}_{p}g$

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The matrices are given as follows: However, I am not able to prove, how to get them

$A=\left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ -{a}_{0}& -{a}_{1}& -{a}_{2}& \cdots & -{a}_{n-1}\end{array}\right]$

$C=\left[\begin{array}{ccccc}1& {b}_{1}/{b}_{0}& {b}_{2}/{b}_{0}& \cdots & {b}_{n-1}/{b}_{0}\end{array}\right]$

How do I get the above matrices from the differential equation form as shown above?

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