Final answer:
To find the Laplace Transform using the convolution theorem, we need to find the Laplace transforms of the individual functions and then combine them. For the first problem, the Laplace transform of t²∗teᵗ is 2/(s-1)². For the second problem, the Laplace transform of ∫⁰ₜecosτdτ is 1/(s²+1). For the third problem, the Laplace transform of (s+1)/s[(s+1)²+1] is (s+1)/[(s+1)²+1].
Step-by-step explanation:
To find the Laplace Transform, we can use the convolution theorem. For the first problem, L{t²∗teᵗ}, we can rewrite it as L{t²} * L{teᵗ}, where L{t²} and L{teᵗ} are the Laplace transforms of t² and teᵗ respectively. The Laplace transform of t² is (2!)/s³ = 2/s³, and the Laplace transform of teᵗ is 1/(s-1). Multiplying these two transforms gives us the Laplace transform of t²∗teᵗ which is 2/(s-1)².
For the second problem, L{∫⁰ₜecosτdτ}, we can rewrite it as ∫⁰ₜL{cosτ}dτ, where L{cosτ} is the Laplace transform of cosτ. The Laplace transform of cosτ is s/(s²+1). Integrating this transform from 0 to t gives us ∫⁰ₜL{cosτ}dτ = 1/(s²+1).
For the third problem, L{(s+1)/s[(s+1)²+1]}, we can simplify it by dividing the numerator and denominator by s. This gives us (s+1)/[(s+1)²+1]. Taking the Laplace transform of (s+1)/[(s+1)²+1] gives us the same expression.