1.2.1. Solving Differential Equations

Try these first

1. 1.

   What is the formula for the Laplace Transform of the first derivative?

   Hint

   Recall the components: the transform of the original function and its initial value.

2. 2.

   Define an Initial Value Problem (IVP).

   Hint

   Focus on what needs to be known before solving.

3. 3.

   What is the result of L{f'(t)}?

   • sF(s) + f(0)
   • sF(s) - f(0)
   • sf(0) - F(s)

   Hint

   Think about how derivatives relate to the function itself.

4. 4.

   True or False: The second derivative transforms to L{f''(t)} = s^2F(s) - sf(0)

   • True
   • False

   Hint

   Recall the formula and check for omissions.

5. 5.

   Solve the differential equation y''' + y'' + 2y' + y = 0 using Laplace Transforms.

   Hint

   Identify the characteristic equation formed by the s-domain.

6. 6.

   For the function f(t) = t^2e^(2t), calculate its Laplace Transform directly.

   Hint

   Look up how polynomial functions behave under transforms!