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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].

 

 

π

y

 

 

 

 

 

 

 

 

 

 

 

-3π

π

π

 

 

 

 

 

 

 

 

 

 


 

 

 

 

(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 ?