1. Find the inverse Laplace transform of the following...

1. Find the inverse Laplace transform of the following functions: a. F(s) = ((s - 2)(s - 4)) / ((s - 1)(s - 3)(s - 5)) b. F(s) = s^2 / (s^4 - 16) 2. Use the Laplace Transform to solve the IVP: y'' + 4y' - 5y = e^2t, y(0) = y'(0) = 0

Transcript

00:02 Hello students we need to find inverse laplace transformation of the given function that is f of s is equal to s minus 2 into s minus 4 divided by s minus 1 into s minus 3 into s minus 5 partial fractions of s minus 2 into s minus 4 divided by s minus 1 into s minus 3 into s minus 5 is given by 3 by 3 by 3 by 3 by 3 by 3 by 3 by 3 by 3 8 into s minus 1 plus 1 by 4 into s minus 3 plus 3 by 8 into s minus 5 now applying inverse laplace transformation on both the side we get 3 by 8 inverse laplace of 1 by s minus 1 plus 1 by 4 into inverse laplace of 1 by s minus 3 plus 3 by 8 into inverse laplace of 1 by s minus 5 now applying the transformation therefore we get the value 3 by 8 into e rise to the part t plus 1 by 4 into e rise to the power 3 t plus 3 by 8 into e rise to the par 5 t.

01:15 Next we have the function f of s s is equal to s squared divided by s rise to the power 4 minus 16.

01:23 Now taking the partial fractions of s square by s rise to the power 4 minus 16 we get the value 1 divided by 2 into 2.

01:31 S square plus 4 minus 1 by 8 into s plus 2 plus 1 by 8 into s minus 2.

01:40 Now applying the inverse laplace on both the side we get 1 by 2 into let multiply and divide by 2.

01:47 So we have laplace inverse of 2 by s square plus 4 can be written as 2 minus 1 by 8 into inverse laplace of 1 by s plus 2 plus 2 plus 1 by 8 inverse laplace of 1 by s minus 2.

02:02 Now applying the transformation therefore we get the value 1 by 4 into sine 2 t minus 1 by 8 into e rise to the par 2 t plus 1 by 8 into e rise to the power 2 t.

02:18 Next we need to solve using laplace transformation y double dash plus 4 y dash minus 5 y is equal to e rise to the power 2 t given y of 0 is equal to y dash of 0 is equal to 0 the initial value.

02:34 Condition...

Question

1. Find the inverse Laplace transform of the following functions: a. F(s) = ((s - 2)(s - 4)) / ((s - 1)(s - 3)(s - 5)) b. F(s) = s^2 / (s^4 - 16) 2. Use the Laplace Transform to solve the IVP: y'' + 4y' - 5y = e^2t, y(0) = y'(0) = 0

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