take inverse laplace
use inverse laplace rule
so here
and
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take inverse laplace
apply laplace rule
here we have already find inverse laplace of
and that is
but as explained above here a=2, so inverse laplace is
1. (a) Using the Tables of Laplace transforms, along with the operational theorems, de- termine the...
1. (a) Using the Tables of Laplace transforms, along with the operational theorems, de- termine the inverse Laplace transform of s +3 82 6s 16 (b) Hence deduce the inverse Laplace transform of 83 -6s e s2 6s 16 1. (a) Using the Tables of Laplace transforms, along with the operational theorems, de- termine the inverse Laplace transform of s +3 82 6s 16 (b) Hence deduce the inverse Laplace transform of 83 -6s e s2 6s 16
8. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of 1 12 8iw w2 _ (b) Hence, determine the inverse Fourier transform of -iw 12 8iw -w2' (c) Use Fourier transforms to solve d2y ,dy + dt2 12y (t 1 8 8. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of 1 12 8iw w2 _ (b) Hence, determine the...
8. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the Fourier transform of sgn(t 1)e4t-1| (b) Hence, find the Fourier transform of 5i sgn(t 1) eit-4e-1| (Simplify your answer. 8. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the Fourier transform of sgn(t 1)e4t-1| (b) Hence, find the Fourier transform of 5i sgn(t 1) eit-4e-1| (Simplify your answer.
9. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of 4 9+w2 9w2 (b) The function f(t) satisfies the integral equation -4u f(t u) H(u) du 6sgn(t)e-3¢|. f(t) 4 Find the Fourier transform of the function f(t) and hence find the solution f(t) 9. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of 4 9+w2 9w2 (b) The function f(t)...
Answers are: 9. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of iw 4 + 9 w2 9 w2 (b) The function f(t) satisfies the integral equation: OO -4u Н(u) du + 6sgn(t) е З, f(t) 0- ft - u) е" = 4 e -OO Find the Fourier transform of the function f(t) and hence find the solution f(t) 7 "(1-)н, (b) Transform the equation by using the convolution Theorem:...
9. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of iw 45iw(iw2 (b) The function f(t) satisfies the integral equation: f(t)2 f(t - u) sgn(u) du — 6е" H(). Find the Fourier transform of the function f (t) and hence find the solution f(t) The sign function sgn(t) = 1 if t 0, 0 if t 0 and -1 if t < 0 H(t) is the Heaviside unit step function...
6. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the Fourier transform of -3t e sin (2(t5) H(t5) (b) Hence, find the Fourier transform of 6 e-3t-it sin (2(t +5)) H(t+5). 6. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the Fourier transform of -3t e sin (2(t5) H(t5) (b) Hence, find the Fourier transform of 6 e-3t-it sin (2(t +5)) H(t+5).
Answers are: 10. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of 40 w2 - 13iw (b) Use Fourier transforms to solve dy -5t + 8y — 9е эH (). dt 15t H(t) 1 8t (а) (1) Н, 9 Then solve for 5iw (b) Apply the Formula of transform of derivatives to get: (iw+8)Y(w) Y (w) and take the inverse transform to have -8t у(0) — Зе 5 н(t) —...
7. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the Fourier transform of -3t e H(t 4) (b) Hence find the Fourier transform of -6 е б-3)е н(t — 4)
#2 (50 pts) Find the inverse Laplace transform of the following function by using Theorems and tables 2s +5 $? +68 +34)