Differential Equations

An equation in the form of dy/dx = f(x) g(y) is called a differential equation of the first order, where f(x) and g(y) are functions of x and y respectively

E.g.

dy/dx = xy is a differential equation where x and y are functions of x and y respectively.

In order to integrate a differential equation, the variables must be separated as follows:

dy/dx = f(x) g(y)
dy/g(y) = f(x) dx

E.g.1

dy/dx = xy
∫ dy/y = ∫ x dx
ln y = x2/2 + c
y = ex2 + c

E.g.2

dy/dx = cos2y ex
dy/cos2y = ex dx
∫ dy/cos2y = ∫ ex dx
∫ dy sec2y = ∫ ex dx
tan y = ex + c

Exponential Growth / Decay

If a rate of change is proportional to its quantity, such a rate is called Exponential Growth / Decay.

E.g.1

The rate of increase in population of a colony of bacteria is proportional to the number of bacteria in the colony at a given time. Therefore, such a rate of increase is an exponential growth.

So, dN/dt ∝ N
dN/dt = k N
dN/N = k dt
∫ dN/N = ∫ k dt
ln N = kt + c
N = ekt + c
N = ekt X ec ---- 1
Let N = N0 when t = 0
N0 = ec
Sub this in 1
N = N0ekt
This is exponential growth. The following image indicates the graphical nature of the growth.

Exponential growth

E.g.2

The rate of decay of radioactive nuclei in a radioactive substance is directly proportional to the number of remaining nuclei at a given time. Therefore, this is an exponential decay.

So, dN/dt ∝ -N ---- the negative sign indicates a decay or a loss
dN/dt = -k N
dN/N = -k dt
∫ dN/N = ∫ k dt
ln N = -kt + c
N = e-kt + c
N = e-kt X ec ---- 1
Let N = N0 when t = 0
N0 = ec
Sub this in 1
N = N0e-kt
This is exponential decay. The following image indicates the graphical nature of the decay against time.

Exponential Decay

 

 

 

 

Resources at Fingertips

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level. They are organized according to these specific levels.
The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.
This is also an opportunity to pay tribute to the intellectual giants like Newton, Pythagoras and Leibniz, who came up with lots of concepts in maths that we take for granted today - by using them to serve mankind.

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