19. Partial Differential Equations

Try these first

1. 1.

   Define the Laplace Transform.

   Hint

   What does the transform change about the function?

2. 2.

   State the first property of Laplace Transform.

   Hint

   Think about how we can combine functions.

3. 3.

   What does the Laplace Transform do?

   • a) Converts functions into integers
   • b) Converts time functions into complex functions
   • c) Changes differential equations into algebraic equations

   Hint

   Think about the domain shift it creates.

4. 4.

   Laplace Transforms can handle initial conditions naturally. True or False?

   • True
   • False

   Hint

   How do we incorporate initial conditions in other methods?

5. 5.

   Given the heat equation $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$ with initial conditions $u(0, t) = 0$ and $u(x, 0) = f(x)$, solve for $u$ using Laplace transforms.

   Hint

   Break down the steps into manageable parts.

6. 6.

   Consider the wave equation \frac{^2u}{^2t} = c^2 rac{^2u}{^2x} with conditions $u(x, 0) = g(x)$ and $u_t(x, 0) = h(x)$. How would you approach this with Laplace transforms?

   Hint

   Focus on initial conditions integration into the transform.