### 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 = x^{2}/2 + c

y = e^{x2 + c}

**E.g.2**

dy/dx = cos^{2}y e^{x}

dy/cos^{2}y = e^{x} dx

∫ dy/cos^{2}y = ∫ e^{x} dx

∫ dy sec^{2}y = ∫ e^{x} dx

tan y = e^{x} + 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 = e^{kt} + c

N = e^{kt} X e^{c} ---- **1**

Let N = N_{0} when t = 0

N_{0} = e^{c}

Sub this in **1**

N = N_{0}e^{kt}

This is exponential growth. The following image indicates the graphical nature of the 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 e^{c} ---- **1**

Let N = N_{0} when t = 0

N_{0} = e^{c}

Sub this in **1**

N = N_{0}e^{-kt}

This is exponential decay. The following image indicates the graphical nature of the decay against time.