Applications of ode and matrices

Faizan Shabbir 13-ME-032

Shameel Shahid Hashmi 13-ME-158

Ibrahim Munir 13-ME-102

Muhammad Ali Khan 13-ME-071

Danial Zafar Gondal 13-ME-027

Khurram Shahzad 13-ME-059

Exponential Growth & Decay

Radioactive Decay

Falling Object Problem

• Exponential functions come into in situations in which

the rate at which some quantity grows or decays.

• As we are taking about rate of change of some quantity.

Therefore,

𝒅𝒚

𝒅𝒕 ∝ y

And final form is

y = y°ekt

It is known that cells of a given bacterial culture divide

every 3.5 hours (on average). If there are 500 cells in a

dish to begin with. How many will there be after 12

hours?

• The half life of a radioactive element is the time required for half

of the radioactive nuclei present in the sample to decay.

• For the quantity to reduce one half

• According to differential equation

𝒅𝒚

𝒅𝒕 = 𝒌𝒚

Let y0 be the number of radioactive nuclei present initially. Then the

number y of nuclei present at time t will be given by

y = y°ekt

𝒉𝒂𝒍𝒇 𝒍𝒊𝒇𝒆 = 𝒍𝒏 𝟐

−𝒌

1) The number of atoms of plutonium-210 remaining after t

days, with an initial amount of y0 radioactive atoms, is given

by:

y = y°e(-4.95x10-3)t

Find the half-life of plutonium-210.

2) Uranium 237 has a half-life of about 6.78 days. If there are 10

grams of Uranium 237 now, how much will be left after 2

weeks?

If a stationary object at height y0 is dropped and is not pushed with any force. Gravity causes the object to fall and to accelerate as it falls. Air resistance will be ignored. We use the fact that the acceleration due to the downward force of gravity is g = −9.81m/s2 = −32ft/s2. (Note: The acceleration is considered to be negative because the motion of the object is downward.)

We use the following facts:

• Acceleration is the rate of change in velocity over time so,

𝒅𝒗

𝒅𝒕 = 𝒂

Since, with no air resistance, the acceleration of the object is constant

at g, we get the mathematical model

𝑑𝑣

𝑑𝑡= g

• Velocity is the rate of change in position over time, so

𝑣 =𝑑𝑦

𝑑𝑡

V (0) = 0 , y (0) = yo

An object is dropped from a height of

500 m. When will the object reach

ground level, and with what speed?

Cryptography

Area of a Triangle

Test for Collinear Points

It involves two processes:

• Encryption Process

• Decryption Process

• Convert the text of the message into a

stream of numerical values.

• Place the data into a matrix.

• Multiply the data by the encoding matrix.

• Convert the matrix into a stream of

numerical values that contains the

encrypted message.

The easiest scheme is to let space=0, A=1, B=2, Y=25, and Z=26.

• Encrypt the word REDRUM by using matrix [B] as encryption.

B =

• The message "Red Rum" would become 18, 5, 4, 0, 18, 21, 13.

4 −2−1 3

• Place the encrypted stream of numbers that

represents an encrypted message into a matrix.

• Multiply by the decoding matrix. The decoding

matrix is the inverse of the encoding matrix.

• Convert the matrix into a stream of numbers.

• Convert the numbers into the text of the original

message.

Perform decryption of following numbers.

67, -21, 16, -8, 51, 27, 52, -26.

Encryption matrix is

B = 4 −2

−1 3

• Find vertices (x1,y1), (x2,y2), and (x3,y3) of triangle in

(X, Y) plane

• Use the formula

Area = ± 1/2

• No matter answer comes positive or negative.

Always take positive value for area.

X1 Y1 1

X2 Y2 1

X3 Y3 1

• Find area of the following triangle.

• Find X & Y coordinates of points

• If the equation

satisfies. It means points are collinear.

x1 y1 1

x2 y2 1 = 0

x3 y3 1

Check if points A,B,C are collinear.

• B = (9,48)

• A = (3,18)

• C = (6,33)

THANK YOU…

ANY QUESTION PLEASE

……???