Question

Using the table of Laplace transforms, find Laplace transforms of the following functions: a) t3sint b) te2t+cos(3t);

c) te2tf(3t) provided that L{f}=F is known.

Answer 1

To find the Laplace transform of a function, we can use the table of Laplace transforms. For each function in the given question, we went through the steps to find their Laplace transforms using the appropriate rules and properties.

Step-by-step explanation:

To find the Laplace transform of a function, we can use the table of Laplace transforms. Let's go through each function:

a) t^3 sin(t): Using the table, the Laplace transform is (6s^2 + 6s + 1) / (s^4 + 2s^2 + 1).

b) te^(2t) + cos(3t): The Laplace transform of te^(2t) is 1 / (s - 2)^2 and the Laplace transform of cos(3t) is s / (s^2 + 9). Combining these transforms, the Laplace transform of the given function is 1 / (s - 2)^2 + s / (s^2 + 9).

c) te^(2t)f(3t): Since L{f(t)} = F(s), we can use the property of linearity to find the Laplace transform of te^(2t)f(3t). The Laplace transform will be e^(2t)F(3s-2).