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The Linearity of the Laplace Transform is a key property making it a powerful tool in analyzing and solving linear systems. In simple terms, linearity means that the Laplace Transform of a sum of functions is equal to the sum of their individual transforms. This property can be mathematically represented as:

• If you have functions \( f(t) \) and \( g(t) \), then \( \mathscr{L}\{af(t) + bg(t)\} = a \mathscr{L}\{f(t)\} + b \mathscr{L}\{g(t)\} \), where \( a \) and \( b \) are constants.

This allows us to break complex expressions into more manageable parts. For the problem of finding the Laplace Transform of \( e^{2t}(t-1)^2 \), linearity lets us split this into three separate components:

• \( \mathscr{L}\{t^2 e^{2t}\} \)
• \( -2 \mathscr{L}\{t e^{2t}\} \)
• \( \mathscr{L}\{e^{2t}\} \)

By handling each part independently, solving for the Laplace Transform becomes significantly simpler.

The Shifted Property for Exponentials is a crucial concept when working with Laplace Transforms involving exponential functions. When you encounter a function in the form of \( e^{at}f(t) \), this property helps you transform it into a simpler form. Mathematically, it states:

• \( \mathscr{L}\{e^{at}f(t)\} = F(s-a) \), where \( F(s) \) is the Laplace Transform of \( f(t) \).

In our problem, we have terms like \( t^2 e^{2t} \), \( t e^{2t} \), and \( e^{2t} \). Applying the shifted property allows us to shift the s-variable in the transform to account for the exponential's effect:

• For \( \mathscr{L}\{t^2 e^{2t}\} \), it becomes \( \frac{2}{(s-2)^3} \).
• For \( \mathscr{L}\{t e^{2t}\} \), it transforms to \( \frac{1}{(s-2)^2} \).
• For \( \mathscr{L}\{e^{2t}\} \), it results in \( \frac{1}{s-2} \).

By shifting \( s \) by \( 2 \) (since \( a=2 \)), each transform becomes much easier to compute, giving us manageable forms.

Understanding the Laplace Transform Definition is crucial before diving into specific properties or calculations. The Laplace Transform of a function \( f(t) \) is given by the following integral:

• \( \mathscr{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt \)

This integral effectively transforms a time-domain function \( f(t) \) into a complex frequency-domain representation \( F(s) \). This transformation makes it easier to solve differential equations and perform system analysis because it turns derivatives into algebraic expressions.
To start with the transformation of \( e^{2t}(t-1)^2 \), we first expanded it into simpler terms. Then, by following the definition, we calculated the Laplace Transform for each of these components:

• \( t^2 e^{2t} \) simplifies to \( \frac{2}{(s-2)^3} \).
• \( t e^{2t} \) becomes \( \frac{1}{(s-2)^2} \).
• \( e^{2t} \) is reduced to \( \frac{1}{s-2} \).

These results show the power of the Laplace Transform in breaking down complex differential expressions into a comprehensible algebraic form.