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Prepared by:-
Name Arvindsai Nair
Dhaval Chavda
Saptak Patel
Abhiraj Rathod
Enrollment no.130454106002
130454106001
140453106015
140453106014
The Laplace TransformThe Laplace Transform
•Suppose that f is a real- or complex-valued function of the (time)variable t > 0 and s is a real or complex parameter. •We define the Laplace transform of f as
The Laplace TransformThe Laplace Transform
•Whenever the limit exists (as a finite number). When it
does, the integral is said to converge.
•If the limit does not exist, the integral is said to diverge
and there is no Laplace transform defined for f .
The Laplace TransformThe Laplace Transform
•The notation L ( f ) will also be used to denote the Laplace
transform of f.
•The symbol L is the Laplace transformation, which acts on
functions f =f (t) and generates a new function,
F(s)=L(f(t))
The Laplace Transform of δ(t – a)To obtain the Laplace transform of δ(t – a), we write
and take the transform
The Laplace Transform of δ(t – a) To take the limit as k → 0, use l’Hôpital’s rule
This suggests defining the transform of δ(t – a) by this limit, that is,
(5)
Inverse of the Laplace TransformIn order to apply the Laplace transform to
physical problems, it is necessary to invoke the inverse transform.
If L(f (t))=F(s), then the inverse Laplace transform is denoted by,
Example : Unrepeated Complex Factors. Damped Forced VibrationsQ.Solve the initial value problem for a damped mass–spring
system, y + 2y + 2y = r(t), r(t) = 10 sin 2t if 0 < t < π and 0 if t > π; y(0) = 1, y(0) = –5.Solution. From Table 6.1, (1), (2) in Sec. 6.2, and the second
shifting theorem in Sec. 6.3, we obtain the subsidiary equation
We collect the Y-terms, (s2 + 2s + 2)Y, take –s + 5 – 2 = –s + 3 to the right, and solve,
(6)
For the last fraction we get from Table 6.1 and the first shifting theorem
(7)
continued
In the first fraction in (6) we have unrepeated complex roots, hence a partial fraction representation
Multiplication by the common denominator gives 20 = (As + B)(s2 + 2s + 2) + (Ms + N)(s2 + 4). We determine A, B, M, N. Equating the coefficients of each
power of s on both sides gives the four equations