2.5.1. Solving Differential Equations

Try these first

1. 1.

   Find the Laplace Transform of f(t) = 2t + 3.

   Hint

   Remember to apply linearity and find the transforms of each part.

2. 2.

   What is the Laplace Transform of f(t) = sin(t)?

   Hint

   Refer to the formula for sine.

3. 3.

   What does the Linearity Property state?

   • Makes functions nonlinear
   • Applies to summations
   • Isolates constants within transforms

   Hint

   Think about how we can handle combinations of functions.

4. 4.

   True or False: The Laplace Transform can only be applied to single functions.

   • True
   • False

   Hint

   Recall how we apply transformations to multiple functions.

5. 5.

   Given f(t) = 5t + 3cos(t) and g(t) = 2t² + sin(2t), find ℒ{f(t) + g(t)}.

   Hint

   Break it down using known transforms!

6. 6.

   Prove the dependency on linearity by applying the Laplace Transform to a third-order equation involving constant coefficients.

   Hint

   Recall definitions of derivatives and transforms.