Obtain the Laplace transform of each of the following functions: (a) e^-2 t cos3 t u(t) (b) e^-2

Obtain the Laplace transform of each of the following...

Key Concepts

Differentiation Property in Laplace Transform

The differentiation property in the Laplace transform domain allows the transformation of time-domain products involving t and another function into derivatives of the Laplace transform with respect to s. For instance, L{t f(t)} = ?dF(s)/ds, where F(s) is the Laplace transform of f(t). This property is particularly useful when dealing with functions multiplied by time, such as t·sin(bt), and simplifies the computation of the Laplace transform.

Hyperbolic Function Transforms

Hyperbolic function transforms involve the Laplace transforms of hyperbolic cosine and sine functions. The expressions, such as L{cosh(bt)} = s/(s² ? b²) and L{sinh(bt)} = b/(s² ? b²), arise from the definitions of hyperbolic functions in terms of exponential functions. These transforms are useful in contexts where the system behavior is analogous to hyperbolic growth or decay.

Trigonometric Function Transforms

Trigonometric function transforms refer to the standard Laplace transforms of sine and cosine functions. For example, L{cos(bt)} = s/(s² + b²) and L{sin(bt)} = b/(s² + b²), where b is a constant. These formulas are fundamental in the analysis of oscillatory systems and in solving differential equations with sinusoidal inputs.

Unit Step Function

The unit step function, denoted by u(t), is used to indicate that a function is defined for t ? 0 and is zero for t < 0. It ensures causality in practical applications, particularly in systems and signal analysis, and is incorporated into the Laplace transform framework to restrict integration to the appropriate domain.

Laplace Transform

The Laplace transform is an integral transform that converts a time-domain function into a complex frequency-domain representation. It is defined by the integral L{f(t)} = ??? e^(?st)f(t) dt, where s is a complex number. This transform is a central tool in solving differential equations, analyzing systems, and simplifying convolutions into products.

Exponential Shifting Theorem

The exponential shifting theorem, also known as the first shifting theorem, states that multiplying a function f(t) by an exponential term e^(?at) results in a shift of the Laplace transform in the s-domain. In particular, if F(s) is the Laplace transform of f(t), then L{e^(?at)f(t)} = F(s + a). This property is crucial for handling functions involving exponential decays.

Transcript

00:01 In this problem of laplace transform we have to find the laplace transform of say first is e to the power minus 2t multiplied with cos so this is multiplied with cost 3t multiplied with ut so using the multiplication formula we can write first the laplace transform of sign so here and here we have cost so we can write the laplace of cost so la plus of cost so la plus of cos a t is given why s divided with s square plus a square but remember here we have multiplied e to the power minus to t so this would be say when we replace it so this would be of the form of s and this would be minus minus plus so this would be s plus and here this would be again same so s plus two whole is square plus a say we have replaced we have to replace a with three so this would be three square is 9 so here we have the laplace of this term say we can say laplace of e to the power minus 2 t pos 3 t multiplied with u t is equals to this is s plus 2 divided with s plus 2 whole square plus 9 now in part b we have to find the laplace of e to the power minus 2 t signed 4 t and and ut and again we know that the laplace of sine a t is given why this is a divided with here this would be s is square plus a square but here we have multiplied e to the power minus 2 t so that means here this would be replaced s is replaced with s plus 2 so now we have to replace s with s plus 2 and we have to replace a with 4 so this would be say we can say here this would be laplace of e to the power minus 2 t sine 4 t u t is equals to the value of age 4 so this would be 4 s plus 2 so this would be s plus 2 whole is square and here age replaced with 4 so this would be 4 square age 16 and now again we have to do the laplace transform of e to the power minus 3 t multiplied with cos hyperbolic 2 t u t this is 2t and multiplied with u t now we have to find the laplace of this term and we know that laplace of cost this is cos hyperbolic a t is given why this is s divided with s s square minus a square so here we have to…

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