Laplace transform practice: For the input forcing function X(t) = 2 t^2 (for t ≥ 0), what is its Laplace transform?

Introduction / Context:
Laplace transforms convert time-domain signals into the s-domain for system analysis and control design. Powers of time are common test inputs for checking transform fluency.

Given Data / Assumptions:

• X(t) = 2 t^2 for t ≥ 0.
• Standard transform: L{t^n} = n! / s^(n+1).

Concept / Approach:
Use the known transform of t^n and scale by constants. For n = 2, L{t^2} = 2! / s^3 = 2 / s^3. Multiplying by the constant 2 gives the final result.

Step-by-Step Solution:
Start with L{t^n} = n! / s^(n+1).Set n = 2 → L{t^2} = 2 / s^3.Multiply by 2 (from 2 t^2) → 2 * (2 / s^3) = 4 / s^3.

Verification / Alternative check:
Differentiate the known result L{t} = 1/s^2 with respect to s (using properties) to recover L{t^2} and confirm.

Why Other Options Are Wrong:

• 2/s^2 or 4/s^2: incorrect power of s; corresponds to L{t} forms.
• 2/s^3: misses the factor of 2 from the input amplitude.
• 2/s^4: corresponds to t^3 behavior.

Common Pitfalls:
Forgetting the constant multiplier or misapplying the exponent in s.

Final Answer:
4 / s^3