Find the inverse Laplace transform of each of the following:...

Find the inverse Laplace transform of each of the following: (a) $F(s)=-\frac{2}{3 s-4}$ (b) $F(s)=\frac{1}{s^{2}-8}$ (c) $F(s)=\frac{3 s-4}{s^{2}+16}$ (d) $F(s)=\frac{7 s^{2}+27}{5^{3}+95}$ (e) $F(s)=\frac{4 s}{\left(s^{2}-1\right)^{2}}$ (f) $F(s)=-\frac{s^{2}-6 s+14}{s^{3}-s^{2}+4 s-4}$

Find the inverse Laplace transform of each of the following:
(a) $F(s)=-\frac{2}{3 s-4}$
(b) $F(s)=\frac{1}{s^{2}-8}$
(c) $F(s)=\frac{3 s-4}{s^{2}+16}$
(d) $F(s)=\frac{7 s^{2}+27}{5^{3}+95}$
(e) $F(s)=\frac{4 s}{\left(s^{2}-1\right)^{2}}$
(f) $F(s)=-\frac{s^{2}-6 s+14}{s^{3}-s^{2}+4 s-4}$

Key Concepts

Inverse Laplace Transform

The inverse Laplace transform is the process of finding a function in the time domain, f(t), whose Laplace transform is the given function F(s). It is a fundamental technique in solving linear differential equations and system dynamics by transforming complex algebraic expressions in the s-domain back into time-domain functions.

Standard Laplace Transform Pairs

Standard Laplace transform pairs are established, commonly recognized correspondences between functions in the time domain and their images in the s-domain. These pairs, which include transforms of exponentials, sine and cosine functions, and polynomials, are used as building blocks for finding inverse transforms by matching forms and applying known relationships.

Partial Fraction Decomposition

Partial fraction decomposition involves breaking down a complex rational function into a sum of simpler fractions whose inverse Laplace transforms are known or easier to compute. This technique is especially useful when the denominator can be factored into linear or irreducible quadratic factors, allowing for systematic inversion by working with each simpler term.

Shifting Theorem

The shifting theorem, also known as the first translation theorem, states that multiplying a time-domain function by an exponential corresponds to a shift in the s-domain. Conversely, a shift in the s-domain is related to the incorporation of an exponential factor in the time domain. This theorem is instrumental in handling transforms that involve terms of the form (s - a) in the denominator or numerator.

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in a form that is easier to associate with standard Laplace transform pairs, particularly those involving sine and cosine functions. By expressing the quadratic in a completed-square format, one can readily identify the corresponding time-domain function, especially when dealing with irreducible quadratic factors.

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