## differential equation

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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