MATH2021 Introduction to Applied Mathematics SEMESTER 1, 2021 EXAMINATIONS
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SEMESTER 1, 2021 EXAMINATIONS
MATH2021
Introduction to Applied Mathematics
1. Consider the second-order non-homogeneous differential equation
y// - 2y/ - 3y = 8xe-x
(a) Find the complementary solution (of the corresponding homogeneous differential
equation).
(b) Use the method of undetermined coefficients to find the general solution of the non-
homogeneous differential equation.
2. Consider the second-order non-homogeneous Cauchy-Euler differential equation x2 - 3x + 4y = x3
(a) Show that the complementary solution (of the corresponding homogeneous differential
equation) is yc (x) = C1 x2 + C2 x2 ln x.
(b) Use variation of parameters to derive the general solution of the non-homogeneous
differential equation.
3. We solve the following second-order differential equation
d2y dy
dx2 dx
using a Taylor series expansion about the ordinary point x = 0.
(a) Show that the coefficients must satisfy the recurrence formula
n(n - 2)an + 4an-2
and determine the free coefficients.
(b) Write out the series solution up to order x4 .
4. Consider the differential equation with 2π-periodic non-homogeneous term
,x, 0 < x < π
y// - 2y = f (x), f (x) = . x2 -π < x < 0 , f (x + 2π) = f (x).
The Fourier series of the non-homogeneous term is given by
f (x) ~ + n ┌ + ┐ + n
n odd n even
(a) Draw the Fourier representation So (x) of f over the interval [-3π, 3π].
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(b) Show that the Fourier sine coefficient of f (x) is indeed
, 4
bn = . π 2n3 , .0,
n odd
n even
[Hint:
x cos(nx) sin(nx)
n n2 ,
x2 sin(nx)dx = 2x sin(nx) + (2 - n2 x2 ) cos(nx) ]
(c) Find a particular solution yp (x) to the differential equation.
5. Consider the linear partial differential equation
∂2u ∂2u ∂2u
∂t2 ∂t∂x ∂x2
(a) Show that its D’Alembert solution is u(x, t) = F (x - t) + G(x + 4t) by letting
u(x, t) = f (x - αt) and deducing two values for α .
(b) Subject to the initial data
u(x, 0) = x2 ,
show that the solution is
u(x, t) = x2 - 12xt - 14t2
6. Consider the initial-boundary value problem
∂u ∂2u ∂u 13
∂t ∂x2 ∂x 4
(a) By separation of variables u(x, t) = X(x)T (t), obtain the following pair of ordinary
differential equations
X// + 3X/ + ╱λ + 、X = 0,
T˙ = -λT.
(b) Find the general solution to the differential equation for X(x) in the case of oscillatory
solutions (λ > -1).
(c) Using the boundary conditions, show that the eigen-values are λn = n2 - 1 with
n = 1, 2, 3, . . . and that the corresponding eigen-functions are Xn (x) = e-3x/2 sin(nx).
(d) Solve the differential equation for T (t).
(e) Write down the general solution to the partial differential equation.
(f) Find the solution to the initial-boundary value problem. What is the value of the
solution u(π/2, t) as t → o ?
2022-05-30