37335 Differential Equations

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37335   Differential Equations

Question 1 (4 + 6 = 10 marks).

(i) The ordinary differential equation

x3y'' − 4xy' + 4y = 0,

has a solution y1(x) = x. Use this to find a second linearly in-dependent solution y2.

(ii) Use variation of parameters to find the general solution of the ODE

y'' − 4y' + 4y = ex.

Question 2 (10 marks).

(a) Obtain the general solution of the ODE

y'' + 4xy' + y = 0,

by letting 

Question 3 (8+2=10 marks).

(i) Look for solutions of the equation

of the form 

Show that the exponents are s1 = −4/3, s2 = −2/3 and that an satisfies

Hence obtain the general solution of the ODE.

(ii) Identify the solution of the equation in (i) in terms of Bessel functions.

Question 4 (7+3=10 marks)

(i) Use the Laplace transform to solve the initial value problem

y'' + 3y' + 2y = e−3x,                              (0.1)

y(0) = 0, y' (0) = 0. Note that we can write

for certain constants A, B, C.

(ii) Obtain the inverse Laplace transform of

as an integral. The transforms you need are in the table at the back of the exam.

Question 5 (3+5+2=10 marks).

(i) Calculate the Fourier sine series for the function

f(x) = 1 − x,   0 ≤ x ≤ 1.

(ii) Consider the following initial and boundary value problem for the heat equation.

By looking for solutions of the heat equation of the form u(x, t) = X(x)T(t) show that the functions X and T must satisfy the problems

X'' (x) = λX(x),   X(0) = X(1) = 0,

and

T' (t) = 4λT(t)

for some constant λ. Explain how the possible values of λ are obtained. Hence obtain an expression for X and T.

(iii) Use your answer for part (i) and part (ii) to write down the so-lution of the given initial and boundary value problem for the heat equation.