differential equation
use the maple program to slov the equation and make sure that the assignment is chose one type of equation so i chose the differential equation
Write a program in Maple ( or Mupad) to use the techniques of Laplace
transforms to solve the system of equations
d2 x
= ax + by
dt2
d2 y
= cx + dy
dt2
for any specified values of a, b, c, d, x(0), y(0), Dx(0), Dy(0). Use examples to
test this program and in particular use it in the case
a = b = c = d = 1, x(0) = 1, Dx(0) = 0, y(0) = 0, Dy(0) = 1
20 Marks
To complete the assignment apply your program to solve ONE of the
two physical systems given below. You should choose appropriate and realistic parameters and illustrate your solutions with both an analytical and
graphical output.
30 Marks
1. The equations for two identical masses coupled by three springs are
d2 x
m 2 = -kx – k(x – y)
dt
d2 y
= -k(y – x) – ky
dt2
where x and y are the displacements of the masses from their equilibrium position.Initially the masses are displaced from their equilibrium
positions and then released so that
m
x(0) = a, y(0) = b, Dx(0) = Dy(0) = 0
Repeat the calculations with different , small displacements, a, b , different realistic force constants and comment on the output.
OR
1
2. The equations for a circuit comprising of two identical LC circuits coupled with a different capacitor ,Cˆ are
d2 I1
= -?(1 + a)I1 – ? 2 aI2
dt2
d2 I2
= -?(1 + a)I2 – ? 2 aI1
dt2
where I1 , I2 are the currents in the two circuits and
?=
1
,
Lc
a=
C
Cˆ
For given, realistic values of L, C, Cˆ solve this system for different initial
currents I1 (0), I2 (0) and derivatives I10 (0), I20 (0).
The figures below illustrate the physical systems.
2
x
y
m
m
k
k
k
md 2 x/dx^2 = -kx – k(x-y), md 2 y/dy^2 = -k(y-x) -ky . Here x and y are the displacements measured
from the masses m and k is the common spring constant.
L
00000000
C1
L
00000000
C3
C2
d^2x/dt^2= -w^2(1+a)x -w^2a y, d^2y/dt^2 = -w^2(1+a)y – w^ax, a= C2/C1, C1=C3, w^2 = 1/(LC1)
Here x an y are the current in the left circuit and right circuit respectively. The C’s are the capacitances and
L an inductance.
3
Figure 1: Spring and Circuit
Differential Equations
Simulation of Daily Temperature Variation
The internal temperature T (t ) of a building is governed by Newton’s law of
cooling and is modelled by the following differential equation:
dT
? ?0.2(T ? A(t )), T (0) ? T0 .
dt
Here t is time (in hours), and A(t ) is the ambient (external) temperature given
2?t
by A(t ) ? A0 ? A1 cos
.
24
(a) Choose appropriate values for the constants T0 , A0 and A1 . Find the
exact solution to the equation (using Maple’s dsolve command – see class
notes) and plot it for a time period of three days.
[5 marks]
(b) Generate approximate solutions to the equation for a step length of 2 hrs
using both the Euler method and the Modified Euler method for a period of
three days, and plot them against the exact solution.
[15 marks]
(c) Suppose t ? 0 refers to mid-day on a Monday and a prediction of the
temperature is required for 4am the following Thursday morning.
Investigate the errors produced by the two numerical methods at this time
(compared with the exact solution) when step lengths of 2 hrs, 1 hr and 30
mins are used. Use your values to complete the table below and
comment.
Steplength
h
Exact
Temp
Euler method
Temp
|Error|
Modified Euler method
Temp
|Error|
2
1
0.5
[10 marks]
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