18. Laplace Transforms & Applications

Try these first

1. 1.

   What is the definition of a Laplace Transform?

   Hint

   Look for the integral representation.

2. 2.

   What are initial conditions, and why are they important in ODEs?

   Hint

   Think about what values you need to begin solving.

3. 3.

   What is the primary purpose of using Laplace Transforms?

   • To convert time functions to algebraic equations
   • To graph functions
   • To derive new functions

   Hint

   Consider what advantage Laplace Transforms provide.

4. 4.

   True or False: Initial conditions can be skipped when using Laplace Transforms.

   • True
   • False

   Hint

   Think about the role of conditions in solving ODEs.

5. 5.

   Given $\frac{d^2y}{dt^2} + 4y = 5$ with initial conditions $y(0) = 1$ and $y'(0) = 0$, apply Laplace Transforms to solve.

   Hint

   Consider what changes with the presence of initial conditions.

6. 6.

   An RLC circuit has $L=2, R=3$. Write its ODE, apply Laplace Transforms, and find the current function using a step input.

   Hint

   Relate circuit parameters to their roles in the differential equation.