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Step, Delta, Convolution and Laplace-Differential Equations-Exam Paper, Exams of Differential Equations

Institute of Mathematics and Applications Differential Equations

Differentiation Equations course is one of basic course of science study. Its part of Mathematics, Computer Science, Physics, Engineering. This is past exam. It helps to prepare in coming paper. It includes: Phase, Lines, Equilibrium, Linear, Differential, Equation, Aquifier, Leak, Isoclines, Plane, Slope, Pseudoperiod

Typology: Exams

2011/2012

Uploaded on 08/07/2012

puja 🇮🇳

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u(t)

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t < −a

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t > a

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��

�� u(t) �� t < −a

�� t > a� �� a ��

�� t = 0 ��

t

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δ(τ − a) dτ = u(t − a)� a

� �� f (t) ��

�� f

(n)

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(n)

(a+)

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�� f (t) �� f

� (t)� ��

�� f r

�

(t) �� f (t) ��

f s

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c

f

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c+

f

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�� p(D)x = q(t)� ��

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�� p(D) �� w(t)

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s

n

- a 0

a n

= 0� � �� x �� p(D)x = 0 �� x(0) = · · · = x

(n−2) (0) = 0 �� x

(n−1) (0) =

a

1

n

�� w(t) = u(t)x(t)� �� p(D)v = u(t) ��

�� t > 0 �� p(D)x = 1 ��

x(0) = = x

(n−1)

· · · (0) = 0 � �� u˙ (t) = δ(t)� v˙(t) = w(t)�

�� p(D)x = q(t) �� x(t) = w(t) ∗ q(t)�

t

�� f (t) ∗ g(t) =

f (t − τ )g(τ ) dτ � 0

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(f ∗ g) ∗ h� f ∗ (ag + bh) = a(f ∗ g) + b(f ∗ h)� f ∗ g = g ∗ f � �� f (t) ∗ δ(t) = f (t)� ��

�� g(t)� ��

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f (t) ∗ g(t)�

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(t) ��

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�� L[w(t)] =

1

� W (s) = L[w(t)] �� p(D)� p(s)

�� p(D)x = f (t) ��

X(s) = W (s)F (s)� �� L[f (t) ∗ g(t)] = F (s)G(s)�

�� f (t) �� g(t) �� f (a−) = g(a−) �� f (a+) =

g(a+) �� a ≥ 0 �

�� F (s) ��

�� f (t)� ��

�� s��

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F (s) = L[f (t)] ��

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t�� a + iω ��

�� e

at �� e

at � �� ω�

�� p(D)z =

e

rt �� z p

= W (r)e

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rt

��

�� |W (iω)| ��

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�� ω� x¨ + 4x = sq(ωt)�

�� ω ��

�� f (t) =

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⎪ ⎨

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0 �� t < 0

1 − t �� 0 < t < 1

t > 1

�� f (t)�

�� f

� (t)�

�� f

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(t) ��

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6 �

�� p(D) �� w(t) = 2u(t)te

−t

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�� x¨ + 2 ˙x + 2x = 1

2 s

�� f (t) �� F (s) = �

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2

2s + 5)

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2

2s + 5)

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s = − 3 ± 4 i ��

� �

� � � �

� �

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4 sin(3ωt) sin(5ωt)

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π 3 5

sin(ωt)

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= A �

4 − ω

2

4 sin(ωt) sin(3ωt) sin(5ωt)

x p

π 4 − ω

2 3(4 − 9 ω

2 ) 5(4 − 25 ω

2 )

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t t

τ

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t

8

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6

=

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(tτ

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�

t

� t

= t

8

0 0 0

t t (^) � � t

�� x(t) = w(t) ∗ e

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e

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−t

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−t

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2

= t

2

e

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0

0 0

�� s

2

X + 2sX + 2X = � �� X = �

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2s + 2)

2 s a b(s + 1) + c

(s + 1)(s

2

2s + 5) s + 1 (s + 1)

2

4

1

�� a =

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2(−1 + 2i)

= 2 + i �� b = 2

2

2(−1) + 5 2 2 i

c = 2� �� f (t) = − 2

1

e

−t

e

−t

2

1

cos(2t) + sin(2t) �

�� s = − 1 �� s = − 1 ± 2 i�

�� e

2 t

e

− 3 t sin(4t)� ��

Step, Delta, Convolution and Laplace-Differential Equations-Exam Paper, Exams of Differential Equations

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Download Step, Delta, Convolution and Laplace-Differential Equations-Exam Paper and more Exams Differential Equations in PDF only on Docsity!

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