1.2.1.1. Example Problems

Try these first

1. 1.

   Find L{t^2}.

   Hint

   Use the formula L{t^n}.

2. 2.

   What is L{1}?

   Hint

   Use the basic transformation for constant function.

3. 3.

   What is the formula for L{f'(t)}?

   • L{f'(t)} = sF(s) - f(0)
   • L{f'(t)} = s²F(s) - f(0)
   • L{f'(t)} = sF(s) + f(0)

   Hint

   Think about how initial values affect the transform.

4. 4.

   True or False: The Laplace Transform can simplify the process of solving differential equations.

   • True
   • False

   Hint

   Review how Laplace transforms are used in various applications.

5. 5.

   Find the Laplace Transform of y''' + 4y'' + 3y' = 0 with y(0) = 1, y'(0) = 0, y''(0) = 2.

   Hint

   Don't forget to express the initial conditions with their correct derivatives.

6. 6.

   Prove L{f(n)(t)} formula from the base principles of Laplace transformations.

   Hint

   Systematically apply the transformations step by step, checking the contribution of each term carefully.