Find the solution of the boundary value problem - use the

Question

Question 1.(a) Let n > -1. Use integration by parts to express

I_n+1 = ₀∫^∞(xⁿe^-x)dx

in terms of I_n.

(Hint: For any n, l'Hospital's rule gives that xⁿ/e^x → 0 as x →∞.)

(b) Hence evaluate the definite integral

₀∫^∞(x^5/2e^-x)dx

given that

I_1/2 = ₀∫^∞(x^-1/2e^-x)dx = √Π

(c) = Using the result in part (a), demonstrate that I_n+1 = n! for any n ∈ {1, 2, 3, 4, . . .}.

(d) In fact, m-factorial for any m can be defined as m! = I_m+1 (you are not required to show this identity). Use this identity, part (a) and part (b) to find the value of (-3/2)!

Question 2. (a)Find the solution of the boundary value problem

x.dy/dx - y = x, y(1) = -1

Write the final solution in the form y = f (x).

(b)Find the solution of the boundary value problem

dy/dx - y/x = tan(y/x), y(2) = π

Write the final solution in the form y = f (x).

Hint: Use a substitution first to simplify the differential equation.

Question 3.(a) Use the formal deftnition of the Laplace Transform to find the Laplace transform of the function

          0 for 0 ≤ t < 3
f (t) =
          2 for t ≥ 3

Make sure mathematical rigour is applied, and key steps are clearly explained.

(b) Show that

L^-1 {1/(s² + a²)² = 1/2a³(sin(at - atcos(at))

for constant a.

Hint: Start with

F(s) =1/(s² + a²) and G(s) = 1/(s² + a²)

(c) Use Laplace transforms to solve the initial value problem

d²y/dt² + 4y = sin(2t), y(0) = 1,dy/dt|₀ = 0 .

Show all working and clearly state each Laplace transform property/rule used.

Note: When stating each Laplace transform property/rule you can refer to the row number in the Laplace transform table on the formulae sheet attached to the assignment, for example: "First we apply the Laplace transform to the function f (t) by using [LT0]."

Hint: The result in part (b) may be useful.