Laplace Transform Exercises
Solutions

Find the Laplace transforms

a) \( 3 e^{-4t} \)

\[ F(s) = \mathcal{L}\{ 3 e^{-4t} \} = 3 \mathcal{L}\{ e^{-4t} \} = 3 \left( \frac{1}{s + 4} \right) \]

\[ \boxed{ \frac{3}{s + 4} } \]

b) \( 2t^2 \)

\[ F(s) = 2 \mathcal{L}\{ t^2 \} = 2 \cdot \frac{2!}{s^{2 + 1}} \]

\[ \boxed{ \frac{4}{s^3} } \]

c) \( 4 \cos(5t) \)

\[ F(s) = 4 \mathcal{L}\{ \cos(5t) \} = 4 \left( \frac{s}{s^2 + 25} \right) \]

\[ \boxed{ \frac{4s}{s^2 + 25} } \]

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