The linearity property of Laplace transforms states that the Laplace transform of a linear combination of functions is the linear combination of their individual Laplace transforms. Mathematically, if $L{f(t)} = F(s)$ and $L{g(t)} = G(s)$, then $L{af(t) + bg(t)} = aF(s) + bG(s)$, where $a$ and $b$ are constants. This property significantly simplifies the process of finding Laplace transforms for complex functions.
Basic Laplace transforms involve knowing the transforms of simple functions like $t^n$, $e^{at}$, $sin(at)$, and $cos(at)$. These form the building blocks for more complex transforms, and using the linearity property lets us break down problems into manageable parts. Understanding both the linearity property and basic transforms is key to solving differential equations and analyzing systems.
Match the term with its correct definition:
| Term | Definition |
|---|---|
| 1. Laplace Transform | A. A property allowing distribution of the transform over sums. |
| 2. Linearity Property | B. A mathematical operator that transforms a function of time to a function of complex frequency. |
| 3. $e^{at}$ | C. $\frac{1}{s-a}$ |
| 4. $sin(at)$ | D. $\frac{a}{s^2 + a^2}$ |
| 5. $cos(at)$ | E. $\frac{s}{s^2 + a^2}$ |
Complete the following paragraph with the correct terms:
The Laplace transform converts a function from the _______ domain to the _______ domain. The _______ property states that $L{af(t) + bg(t)} = aL{f(t)} + bL{g(t)}$, where $a$ and $b$ are _______. The Laplace transform of $sin(at)$ is _______. Applying these concepts simplifies solving _______ equations.
Explain, using an example, how the linearity property simplifies finding the Laplace transform of $f(t) = 3t^2 + 2e^{5t}$.
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