Obtain the transfer function of the system if $y(t)={e}^{-t}-2{e}^{-2t}+{e}^{-3t}\text{and}x(t)={e}^{-0.5t}$

smileycellist2
2020-11-26
Answered

Obtain the transfer function of the system if $y(t)={e}^{-t}-2{e}^{-2t}+{e}^{-3t}\text{and}x(t)={e}^{-0.5t}$

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lobeflepnoumni

Answered 2020-11-27
Author has **99** answers

Step 1

Laplace transform of${e}^{-at}$ is $\frac{1}{(s+a)}$ Thus, Laplace transform

$\text{of}y(t)={e}^{-t}-2{e}^{-2t}+{e}^{-3t}\text{is}:$

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

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

$=\frac{(6-6+2)}{(s+1)(s+2)(s+3)}$

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

Step 2

Also, Laplace transform of$x(t)={e}^{-0.5t}\text{is:}$

$X(s)=\frac{1}{(s+0.5)}$

Step 3

Thus transfer function is:$\frac{(Y(s))}{(X(s))}=\frac{\frac{2}{(s+1)(s+2)(s+3)}}{\frac{1}{(s+0.5)}}$

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

Laplace transform of

Step 2

Also, Laplace transform of

Step 3

Thus transfer function is:

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Find the inverse Laplace transform of the following transfer function:

$\frac{Y\left(s\right)}{U\left(s\right)}=\frac{50}{{(s+7)}^{2}+25}$

Select one:

a)$f\left(t\right)=10{e}^{-7t}\mathrm{sin}\left(5t\right)$

b)$f\left(t\right)=10{e}^{7t}\mathrm{sin}\left(5t\right)$

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Select one:

a)

b)

c)

d)

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asked 2020-11-29

Find the Laplace transform of $f\left(t\right)=t{e}^{-t}\mathrm{sin}\left(2t\right)$

Then you obtain$F\left(s\right)=\frac{4s+a}{{({(s+1)}^{2}+4)}^{2}}$

Please type in a = ?

Then you obtain

Please type in a = ?

asked 2022-01-18

I have the following equation

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and I am told it is separable, but not knowing how that is, I went ahead and solved it using the Exact method.

Let$M=x{y}^{2}+x\text{}\text{and}\text{}N=y{x}^{2}+y$

$My=2xy\text{}\text{and}\text{}Nx=2xy$

$\int M.dx\Rightarrow \int x{y}^{2}+x={x}^{2}{y}^{2}+\frac{{x}^{2}}{2}+g\left(y\right)$

Partial of$({x}^{2}{y}^{2}+\frac{{x}^{2}}{2}+g\left(y\right))\Rightarrow x{y}^{2}+{g\left(y\right)}^{\prime}$

${g\left(y\right)}^{\prime}=y$

$g\left(y\right)=\frac{{y}^{2}}{2}$

the general solution then is

$C={x}^{2}\frac{{y}^{2}}{2}+\frac{{x}^{2}}{2}+\frac{{y}^{2}}{2}$

Is this solution the same I would get if I had taken the Separate Equations route?

and I am told it is separable, but not knowing how that is, I went ahead and solved it using the Exact method.

Let

Partial of

the general solution then is

Is this solution the same I would get if I had taken the Separate Equations route?

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Solve both

a)using the integral definition , find the convolution

$f\ast g\text{of}f(t)=\mathrm{cos}2t,g(t)={e}^{t}$

b) Using above answer , find the Laplace Transform of f*g

a)using the integral definition , find the convolution

b) Using above answer , find the Laplace Transform of f*g