7.3. Laplace Transform: Basic Definition

Try these first

1. 1.

   What is the formula for the Laplace Transform?

   Hint

   Look for the integral definition of the Laplace Transform.

2. 2.

   What does differentiating a function mean?

   Hint

   Think about how slopes of curves are determined.

3. 3.

   What is the general formula for the Laplace Transform?

   • L{f(t)} = ∫[0,∞] e^(-st) f(t) dt
   • L{f(t)} = ∫[0,∞] e^(st) f(t) dt
   • L{f(t)} = ∫[0,∞] f(t)e^(st) dt

   Hint

   Think about the integral form involving an exponential decay.

4. 4.

   True or False: The Multiplication by tn Property results in differentiation in the s-domain.

   • True
   • False

   Hint

   Recall how differentiation affects functions in calculus.

5. 5.

   Prove the multiplication by t^n property for n=2 by calculation, starting from L{f(t)}.

   Hint

   Pay attention to the orders of differentiation and exponential decay in your integral expressions.

6. 6.

   Using the properties of Laplace Transforms, solve for L{e^{2t}sin(3t)}. Express in terms of s.

   Hint

   Identify the base transformation of e^{2t} and use it effectively!