What's the inverse transform of the following.
5/(s-4)3
To find the inverse transform of the given function, we need to use the inverse Laplace transform. The inverse Laplace transform of 5/(s-4)^3 can be found using the concept of partial fractions.
Step 1: Rewrite the function in a partial fraction form
To do this, we express 5/(s-4)^3 as a sum of fractions with simpler denominators. In this case, we need to decompose it into three partial fractions.
1. 5/(s-4)^3 = A/(s-4) + B/(s-4)^2 + C/(s-4)^3
Step 2: Determine the values of A, B, and C
To find the values of A, B, and C, we multiply both sides of the equation by (s-4)^3 and then substitute suitable values for 's' to solve for the unknowns.
5 = A(s-4)^2 + B(s-4) + C
Now, we can choose 's' values that will cancel out two of the terms in the equation. For example, let's substitute s = 4.
5 = A(4-4)^2 + B(4-4) + C
5 = A * 0 + B * 0 + C
5 = C
Therefore, C = 5.
Next, we can differentiate both sides of the equation and substitute s = 4 again.
0 = 2A(4-4) + B
0 = B
Hence, B = 0.
Finally, let's substitute s = 4 into the original equation.
5 = A(4-4)^2 + 0(4-4) + 5
5 = A * 0 + 0 + 5
5 = 5
Since both sides of the equation are equal, there is no restriction on A. We can choose any value for A.
Step 3: Write the partial fraction decomposition
Using the values we found, the partial fraction decomposition is:
5/(s-4)^3 = A/(s-4) + B/(s-4)^2 + C/(s-4)^3
5/(s-4)^3 = A/(s-4) + 0/(s-4)^2 + 5/(s-4)^3
5/(s-4)^3 = A/(s-4) + 5/(s-4)^3
Step 4: Take the inverse Laplace transform of each term
Now that we have the function in partial fraction form, we can take the inverse Laplace transform of each term individually. The inverse Laplace transform of 1/(s-a) is e^(at), where 'a' is a constant. Therefore:
Inverse Laplace transform of A/(s-4) = A * e^(4t)
Inverse Laplace transform of 5/(s-4)^3 = 5 * t^2 * e^(4t)
So, the inverse transform of 5/(s-4)^3 is A * e^(4t) + 5 * t^2 * e^(4t), where A can be any constant value.