Question

Find the inverse Laplace transform of the following functions:

1) F(s) = 1/(s+2)(s+3)
2) F(s) = K/s(s+1)²
3) F(s) = s+4/(s+1)²(s+2)
4) F(s) = 1/s²+2s+5
5) F(s) = s+1/s(s²+s+1)
6)F(s) = 1/s²(s²+ω²)

Answer 1

To find the inverse Laplace transforms for the given functions, partial fraction decomposition, table lookup for known transforms, completing the square, and properties of Laplace transform like frequency shifting are used.

Step-by-step explanation:

To solve the given problems, we need to find the inverse Laplace transform of each function. The Laplace transform is a widely used integral transform in mathematics with applications in engineering and physics. The inverse Laplace transform is used to revert back to the time domain from the frequency domain. Each function requires a different approach:

For functions with polynomial denominators such as 1/(s+2)(s+3) and K/s(s+1)², partial fraction decomposition is typically used followed by the use of known inverse Laplace transforms.For functions like s+4/(s+1)²(s+2), we would similarly use partial fraction decomposition and then apply inverse Laplace transforms to each term individually.Functions like 1/s²+2s+5 resemble the inverse transform of a second-degree polynomial in the denominator, potentially needing completing the square to find a standard form.The function s+1/s(s²+s+1) might be approached by separating it into simpler fractions and identifying inverse transforms associated with quadratic factors or using convolution For expressions like 1/s²(s²+ω²), we look for standard inverse laplace transforms involving squared arguments and possibly trigonometric functions.General strategy will involve looking up table values for known inverse transforms, using convolution, partial fraction decomposition, and properties of the Laplace transform such as frequency shifting.