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Laplace Transformations and Inverse Laplace Transformations

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1. Find the Laplace transform of 6 cos t + 2e −3t .

2. Find the Laplace transform of 2 cosh t + 2t 3.

3. Find the Laplace transform of 2te −3t.

4. Find the inverse Laplace transform of s/s+2.

5. Find the inverse Laplace transform of 1/s+5.

6. Find the inverse Laplace transform of 1/( s + 5 ) ( s 2 + 1).

7. The voltage and current in the Laplace domain for a certain system is given by V (s) = 1/12 1/s^2 +3, I(s) = s/s^2+3. Is this system best described as resistive, capacitive, or inductive?

8. The voltage and current in the Laplace domain for a certain system is given by V(s) = 9 1/s^2+16, I(s) = 1/s^2+16. Is this system best described as resistive, capacitive, or inductive?

9. For a second order RLC circuit, in the s-domain s^2 + R/L s + 1/LC = 0. If R = 2 Ω, C = 2 F, and L = 16 H , is the system damped? If so, how? Characterize the behavior of the system.

10. A resistor and inductor are connected in series with a voltage source v ( t ) = 20sin 30t .If R = 12 Ω, L = 4 H, find the total response of the current. Assume there is initially no current. Identify the transient solution

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The solution shows in steps for the find Laplace and Inverse Laplace transformations and determines which voltages are resistive, capacitive, or inductive.

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1. Find the Laplace transform of 6 cos t + 2e −3t .

We know that Laplace transform of cos at = a/a^2 + s^2 and Laplace transform of exp(at) = 1/s-a.
Using linear property L(aƒ + bg) = aL(ƒ) + bL(g) where a and b are constants.
Then the Laplace transform of 6cost + 2e^-3t = 6L(cost) + 2L(e^-3t) = 6/s^2 + 1 + 2/s + 3

2. Find the Laplace transform of 2 cosh t + 2t 3.

See attached.

3. Find the Laplace transform of 2te −3t.
___ ___
We know that L(tƒ(t)) = ...

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