Posted: September 13th, 2017

differential equation ; Write a program in Maple ( or Mupad) to use the techniques of Laplace transforms to solve the system of equations

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

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