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Problems (Linear Systems)

1.)

 
Let $F(j\omega) = e^{-\omega^{2}T^{2}} be a Fourier transformation of $f(x). Determine the inverse Fourier transformation i.e. $f(x) function under the following condition:

 
$
\int_{0}^{\infty}e^{-ax_{0}^{2}}\text{cos}(bx_{0})dx_{0} = \frac{1}{2} \sqrt{\frac{\pi}{a}}e^{-(\frac{b}{4a})^{2}}

Solution

 

2.)

 
Determine the Laplace transformation of the following functions:

$f(x) = e^{-\alpha x}\text{cos}(\omega x), and $f_{1}(x)=e^{-\alpha x}\text{sin}(\omega x)

Solution

 

3.)

 
Determine the inverse Laplace transformation of the following function:

$
F(s) = \frac{s^{2}+3s+2}{(s-1)(s-2)^{3}(s-4)} \quad \quad f(x)=\mathcal{L}^{-1}\left\{F(s)\right\} = ?

Solution

 

4.)

 
Find the solution of the following differential equation!

$
\frac{d^{2}y(x)}{dx^{2}}+4\frac{dy(x)}{dx}+5y(x) = 10,

where $y(0)=\frac{dy(x)}{dx}\Big{|}_{x=0}=0 is the initial condition.

Solution

 

5.)

 
Find the solution of the following linear second ordered differential equation!

$
\frac{d^{2}y(x)}{dx^{2}}+a^{2}y(x)=A\text{cos}(\alpha x),, with the initial conditions $y(0)=y'(0)=0.

Solution

 

6.)

 
Find the solution of the following linear third ordered differential equation!

$
\frac{d^{3}y(x)}{dx^{3}}+5\frac{d^{2}y(x)}{dx^{2}}+8\frac{dy(x)}{dx}+4y(x)=1(x), , with the initial conditions $y''(0)=y'(0)=y(0)=0

Solution

 

7.)

 
Let $                  h(t)=-A(e^{s_1t}- e^{s_2t}) be the step response function of a linear shift invariant system. Determine the system response function if the input function is $                  f(t)=F_0\text{sin}(\omega t+\varphi)!

Solution

 

8.)

 
Two diagrams of a characteristic transfer function (amplitude and phase) are presented below. Shape-preserving transfer is executing within the $                  0<\omega < \omega_0 frequency band. Please, determine the weighting function of the linear invariant system.

Image
Amplitude diagram

 

Image
Phase diagram

 
Solution

 

9.)

 
Step response function of a system is : $                  h(t)= \left [ 2-e^{-5t}+2e^{-10t} \right ] 1(t). Please determine the weighting function of the system!

Solution

 

10.)

 
Transfer characteristic of a system is as follow:
$                  w(j\omega)= \frac{2(1+j\omega)}{4+3j\omega}
Please determine the step response function of the system by the transfer characteristic!

Solution

 

11.)

 
Solve the following partial differential equation $                 \bigtriangledown^2U(x,y,t)= \frac{1}{c^2}\frac{\partial U(x,y,t) }{\partial t} with the following initial conditions
$                  U(x,y,0)=f(x,y)

$                   \begin{vmatrix}
\frac{\partial U(x,y,t)}{\partial t}
\end{vmatrix}_{t=0}=g(x,y)

And $                  \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}|U(x,y,t)|dxdydt<0

Solution


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