Question

Use the "derivative of transforms" formula to evaluate the given Laplace transform. 5. L{tsinht} 6. L{t²cost} 5. 6s²+2/(s²−1)³

Answer 1

To evaluate the Laplace transform of tsinht, we can use the derivative of transforms formula. Similarly, to evaluate the Laplace transform of t^2cost and (6s^2+2)/(s^2-1)^3, we can use the same formula. Applying the formula, we can simplify the expressions and find their Laplace transforms.

Step-by-step explanation:

1. To evaluate the Laplace transform of tsinht, we can use the derivative of transforms formula. Applying the formula, we get L{tsinht} = -d/ds(L{sinht}). The Laplace transform of sinht is s/(s^2-1), so substituting this into the formula gives us L{tsinht} = -d/ds(s/(s^2-1)).
2. Similarly, to evaluate the Laplace transform of t^2cost, we can use the derivative of transforms formula. Applying the formula, we get L{t^2cost} = -d/ds(L{tcost}). The Laplace transform of tcost is (s^2-2)/(s^2+1)^2, so substituting this into the formula gives us L{t^2cost} = -d/ds((s^2-2)/(s^2+1)^2).
3. To evaluate the Laplace transform of (6s^2+2)/(s^2-1)^3, we can use the derivative of transforms formula. Applying the formula, we get L{(6s^2+2)/(s^2-1)^3} = -d/ds(L{(s^2-1)^-3}). The Laplace transform of (s^2-1)^-3 is -1/2(s^2-1)^-2, so substituting this into the formula gives us L{(6s^2+2)/(s^2-1)^3} = -d/ds(-1/2(s^2-1)^-2).