# SMC A System Is Described by The Following Differential Equation Problems

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1. (30 points) A system is described by the following differential equation:
dy
dy
+ 8
dt2
+12y(t) = 2
dt
dt
dr
+5r(t)
dy
y(0) = 0, = 1;
dt
(a) Compute the zero-input response.
(b) Compute the impulse response, h(t).
(c) Determine the zero-state response when the input is r(t) = 4u(t).
Feel free to use Table 2.1 (attached at the end of this exam).
(d) Determine the total response of the system.
Table 2.1: Convolution Table
No. x1(1)
1 x(t)
|x2(1)
811-T)
u(t)
x1(t)* x2(t) = x2(t)*xu(t)
x(t-T)
2 Mult)
1 pir

3 u(t)
4
Mtu(t)
u(t)
e^2u(t)
tu(1)
lei plat
(1)
λι – λα
λι 4 λ.
torruft)
etu(t)
6 tentu(t)
7 Nu(t)
entu(t)
Mtu(t)
şr educa)
errult)
IN
NIIN
22+1(N-k)!
1N4T (1)
ko
8 Mult)
| Mult)
M+N+ (1)
tem fu(t)
12’u(t)
MINI
(M + N + :)!
F431 – +(21-22)redil
– (1)
(-22)
MINI
M+N+leu(1)
(N + M + :)!

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Tags:
mathematics function

Algebraic equations

differential equation

complex math

total response

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