2.5.1. Solving Differential Equations

The Linearity Property states:

If 𝑓(𝑡) and 𝑔(𝑡) are two functions whose Laplace Transforms exist, and 𝑎, 𝑏 are constants, then:

ℒ{𝑎𝑓(𝑡)+ 𝑏𝑔(𝑡)}= 𝑎ℒ{𝑓(𝑡)}+ 𝑏ℒ{𝑔(𝑡)}

This means the Laplace Transform of a linear combination of functions is the same linear combination of their individual Laplace Transforms.

The Linearity Property refers to the ability to separately transform linear combinations of functions. If you have two functions, 𝑓(𝑡) and 𝑔(𝑡), and you multiply them by constants 𝑎 and 𝑏, the Laplace Transform of this entire expression is simply equal to the Laplace Transform of 𝑓(𝑡) multiplied by the constant 𝑎 added to the Laplace Transform of 𝑔(𝑡) multiplied by the constant 𝑏. This makes computations much easier, as you can transform each function separately and then combine the results.

Consider making a fruit smoothie. If you add 2 bananas and 3 berries to your blender, you can think of making the smoothie as finding the blend of these two fruits. Instead of trying to blend both at once, you can first blend the bananas and then the berries, combining the results at the end. The Linear Property allows you to handle complex mixtures easily, just like how separating ingredients simplifies making a smoothie.

Let us consider two functions 𝑓(𝑡) and 𝑔(𝑡) and constants 𝑎 and 𝑏. Then,

ℒ{𝑎𝑓(𝑡)+ 𝑏𝑔(𝑡)}= ∫ 𝑒−𝑠𝑡 [𝑎𝑓(𝑡)+ 𝑏𝑔(𝑡)]𝑑𝑡

= 𝑎∫ 𝑒−𝑠𝑡 𝑓(𝑡) 𝑑𝑡 + 𝑏∫ 𝑒−𝑠𝑡 𝑔(𝑡) 𝑑𝑡

= 𝑎ℒ{𝑓(𝑡)} + 𝑏ℒ{𝑔(𝑡)}

Thus, the linearity property is proved.

To prove the linearity property, we start by applying the Laplace Transform to the expression ℒ{𝑎𝑓(𝑡) + 𝑏𝑔(𝑡)}. This leads us to compute an integral where we can factor out constants 𝑎 and 𝑏 from the integral because they are just coefficients. The integral now separates into two distinct integrals; one for each function 𝑓(𝑡) and 𝑔(𝑡). After solving both integrals separately, we see that this equals 𝑎ℒ{𝑓(𝑡)} + 𝑏ℒ{𝑔(𝑡)}. This shows that combining the functions and their transforms keeps the process simple and systematic.

Think of proving that making a cake with different layers is the same as making each layer separately and then combining them. You start by combining all ingredients for the base and baking it. Once done with the base, you do the same for the frosting. Each layer's effort is kept separate until the end when you combine them for the final product. Similarly, in the proof of the linearity property, each function's Laplace Transform is handled distinctly and combined, confirming the idea behind linearity.

Applications of Linearity Property

1. Solving Differential Equations:
   • Simplifies the Laplace transform of an equation composed of multiple functions.
   • Enables term-wise transformation and back-transformation.
2. Circuit Analysis:
   • Helps in transforming and analyzing multiple sources in a linear circuit (like RLC circuits).
3. Control Systems:
   • Used for analyzing systems with multiple inputs or superimposed signals.
4. Signal Processing:
   • Applies to decomposition of signals into simpler components.

The Linearity Property has widespread applications across various fields. In solving differential equations, it allows for term-by-term transformations which makes problems more manageable. In circuit analysis, it simplifies understanding circuits with multiple sources, especially in RLC circuits where each component affects the total response. Control systems utilize this property for analyzing systems with numerous inputs and responses, while signal processing employs it to break complex signals into their simpler building blocks. Each of these fields benefits from being able to separate and recombine different components easily.

Imagine organizing a massive concert with different bands. Each band plays its music; however, as the event organizer, you want to understand how each band's performance contributes to the whole show. By using the linearity principle, you evaluate each band's performance independently before piecing together an overview of the whole event. This way, you can easily spot which bands work well together and adjust accordingly. Just like this concert, the linearity property in various fields allows us to analyze and piece together complex systems systematically.

Example 1: Find the Laplace Transform of: 𝑓(𝑡) = 3𝑡2 + 5sin(𝑡) Solution: We know:

• ℒ{𝑡2}= 2/𝑠3
• ℒ{sin(𝑡)}= 1/(𝑠2 + 1) Using linearity: ℒ{3𝑡2 + 5sin(𝑡)} = 3⋅(2/𝑠3) + 5⋅(1/(𝑠2 + 1)) = (6/𝑠3) + (5/(𝑠2 + 1))

Example 2: Given: 𝑓(𝑡) = 4𝑒2𝑡 + 7cos(3𝑡) Find ℒ{𝑓(𝑡)} Solution: We know:

• ℒ{𝑒𝑎𝑡} = 1/(𝑠−𝑎)
• ℒ{cos(𝑏𝑡)} = 𝑠/(𝑠2 + 𝑏2) So, ℒ{4𝑒2𝑡 +7cos(3𝑡)}= 4⋅(1/(𝑠−2)) + 7⋅(𝑠/(𝑠2 +9))

These examples demonstrate the power of the Linearity Property in practice. In the first example, we have a polynomial and a sinusoidal function combined. By using known Laplace transform values for 𝑡² and sin(𝑡), we can apply linearity to compute the overall Laplace Transform easily. The second example introduces exponential and cosine functions, each with its known transform formulas. We can isolate each function’s transformation and combine them without any complication. These examples illustrate how complex expressions involving various functions can be approached simply by leveraging linearity.

Consider pulling together a group project with contributions from different team members. Each member presents their findings (like their individual Laplace transforms) using particular methods. When you gather all contributions, you can compile the project based on everyone’s separate input rather than tackling the entire topic as a single entity at once. This piecemeal approach, much like the examples illustrated, shows how separate contributions combine efficaciously to form a cohesive whole.

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