Question 1: Solve the following using Laplace transforms

y" - 7y' + 10 = 0           y'(0) = 24 && y(0) = 6

Question 2: Solve the following using Laplace transforms

1/2 d²y/dt² - 2dy/dt + 2y = e^2t      

dy/dt(0) = 1 && y(0) = 0

Question 3: Solve the following system using Laplace transforms

4y(t)' + 3x(t)' - 2x(t) = -2e^-3t

x(t) + y(t) = 0

y(0) =1/4 && x(0) = 1/3

Question 4:

a) Prove that the following is a solution to the Two-dimensional Wave equation

solution: u = e^{(x + ct)} - e^(-y+ct), Wave Equation: ∂²u/∂t² = c²( ∂²u/∂x + ∂²u/∂y²)

b) Find the relationship between a and b in the following equation that satisfy the Heat equation

solution: u(x,t) = sin(2b/c x + a)e^-at, Heat Equation: ∂u/∂t = c²∂²u/∂x²

Question 5: Temperature distribution in a titanium plate:

a) You are given a square titanium plate with all the edges held at a constant temperature 20°C and there four (red) heat sources at 0°C, placed in the centre of the plate.

i) Sketch the plate, as well as the heat sources and the contour (constant temperature) lines as t → ∞.

ii) What sort of boundary condition is this?

iii) Use MATLAB and simple numerical methods to calculate the temperature at each node assuming the plate is at steady state. Please hand in the code (upload function on AUTonline will be available closer to the due date) and provide a print out version of a plot, showing the temperature at each node, together with the rest of your assignment.

iv) Use MATLAB to plot the isotherms of the steady state plate.

b) Assume the square titanium plate above has all the edges replaced by a non-perfect insulator, instead of being held at 20°C. Answer the below questions if the four centre points are still held at 0°C.

i) Sketch the plate (grid), the heat sources and estimate the temperature at any given point on the plate (grid) as t → ∞. Explain how you came to this answer.

ii) What sort of boundary condition is this?

Question 6: EIGENFUNCTIONS of a vibrating string

You have been given the model of a vibrating string (e.g. a cello string) together with the EIGENFUNCTIONS of the system.

u_n(x, t) = [A₁cos(λ_nt) + A₂sin(λ_nt)]sin(λ_n/c.x)

with A₁ = 2, A₂ = -2

and c = 2

a) Sketch the fifth mode u₅(x, t=0). Indicate the nodes.

b) What is the maximum displacement of the string at t = L/5 for the fifth mode? Explain how you determined that.

c) At what instances in time t will the displacement of the string for the second mode equal zero (u₂ (x, = 0)? Please explain.

d) What is the maximum displacement of the fifth mode? State the position x at which this will happen? Please explain.