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Question 1 (25 marks) a) Solve the following ODE. y'' + y = 1, y(0) = 0, y'(0) = 1 b) Solve the following initial value problem using Laplace transform method. y'' - 4y' + 13y = \delta(t - 3), y(0) = 0, y'(0) = 3 c) Solve the following system of ODEs by let x(t) and y(t) be the two functions of the variable t. $\frac{dx}{dt} = 2x - 3y$ $\frac{dy}{dt} = y - 2x$ $x(0) = 1, y(0) = 2$ Question 2 (25 marks) a) Find the z-transform of the following sequences: i) $\{n + 2\}u[n]$ ii) $\{n^3\}u[n]$ b) Find the inverse z-transform of the following: i) $Z^{-1} \left[ \frac{z|2|\sin 3}{z^2 - (2|2|\cos 3)z + |2|^2} \right]$ ii) $Z^{-1} \left( \frac{z(z^2 + 4z + 1)}{(z - 1)^4} \right)$ iii) $Z^{-1} \left( \frac{z(z + 1^4)}{(z - 1^3)^3} \right)$

          Question 1 (25 marks)
a) Solve the following ODE.
y'' + y = 1, y(0) = 0, y'(0) = 1
b) Solve the following initial value problem using Laplace transform method.
y'' - 4y' + 13y = \delta(t - 3), y(0) = 0, y'(0) = 3
c) Solve the following system of ODEs by let x(t) and y(t) be the two functions of the
variable t.
$\frac{dx}{dt} = 2x - 3y$
$\frac{dy}{dt} = y - 2x$
$x(0) = 1, y(0) = 2$
Question 2 (25 marks)
a) Find the z-transform of the following sequences:
i)  $\{n + 2\}u[n]$
ii)  $\{n^3\}u[n]$
b) Find the inverse z-transform of the following:
i)  $Z^{-1} \left[ \frac{z|2|\sin 3}{z^2 - (2|2|\cos 3)z + |2|^2} \right]$
ii)  $Z^{-1} \left( \frac{z(z^2 + 4z + 1)}{(z - 1)^4} \right)$
iii) $Z^{-1} \left( \frac{z(z + 1^4)}{(z - 1^3)^3} \right)$

$Question 1 (25 marks) a) Solve the following ODE. y” + y = 1, y(0) = 0, y'(0) = 1 b) Solve the following initial value problem using Laplace transform method. y” - 4y' + 13y = δ(t - 3), y(0) = 0, y'(0) = 3 c) Solve the following system of ODEs by let x(t) and y(t) be the two functions of the variable t. (dx)/(dt) = 2x - 3y (dy)/(dt) = y - 2x x(0) = 1, y(0) = 2 Question 2 (25 marks) a) Find the z-transform of the following sequences: i) {n + 2}u[n] ii) {n^3}u[n] b) Find the inverse z-transform of the following: i) Z^-1[ (z|2|sin 3)/(z^2 - (2|2|cos 3)z + |2|^2)] ii) Z^-1( (z(z^2 + 4z + 1))/((z - 1)^4)) iii) Z^-1( (z(z + 1^4))/((z - 1^3)^3))$

Added by Jeffery C.

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