laplace transform Question 9 | laplace transform of f(t) {t^2, (t-1), 7} | laplace question

laplace transform Question 9 | laplace transform of f(t) {t^2 (t-1) 7} | laplace question

🎥 Welcome to our Laplace Transform tutorial! In this video, we're diving into Question 9, where we'll find the Laplace transforms of three different functions:
𝑡^2, 𝑡 − 1, and 7.

🔍 Let's start by understanding the Laplace transform conceptually, which is a powerful tool in solving differential equations. The Laplace transform converts a function of time, 𝑓(𝑡), into a function of a complex variable, 𝑠. This transformation often simplifies the process of solving differential equations, particularly linear constant coefficient differential equations.

📝 Our first function is
𝑓(𝑡) = 𝑡^2. We'll apply the definition of the Laplace transform and integrate from 0 to ∞. Using integration by parts, we'll find the Laplace transform of 𝑡^2.

📈 Next, we'll tackle
𝑓(𝑡) = 𝑡 − 1. Again, applying the Laplace transform definition, we'll integrate to find its transform. Remembering the properties of the Laplace transform, we'll handle the subtraction of 1 accordingly.

➕ Lastly, we have the constant function 𝑓(𝑡) = 7. While seemingly straightforward, finding its Laplace transform involves a crucial understanding of the Laplace transform definition and properties.

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