We Know, differential equations have different applications in real life such as the law of exponential change, radioactivity, the half-life of a radioactive element, carbon dating temperature, and rate of charge electric circuit.

## What is a differential equation?

An equation involving independent and dependent variables and the derivative or differentials of one or more dependent Valuables with respect to one or more independent valuable Called differential equation or **an equation containing derivatives are called Differential equations**

**Examples** for differential equations

- dy / dx = sin x + cos x
- d
^{2}y / dx^{2}+ dy/dx + y = 0 - d
^{2}y / dx^{2}= xy - ∂y / ∂x = y

## There are two main classes of differential equations

**Ordinary differential equations****partial differential equations**

## Ordinary differential equations

Differential equations that involve derivatives with a single indigent variable are known as ordinary differential equations.

from the above example, 1, 2 & 3 are an ordinary differential equation

## Partial differential equations

Differential equations contain two or more independent variables and partial derivatives with respect to them is called partial differentials

Example 4 is a partial differential equation

## Order of a differential equation

The order of the highest order derivative involved in a differential equation is called the order of the differential equation

Examples 1 & 4 are 1st order and 2 & 3 are second-order differential equations

## Degree of a differential equation

The degree of a differential equation is the degree of the highest order derivative of the equation.

The above example, all are first-degree differential equation

(dy/dx)^{4} = xy

(d^{2}y/dx2)4 + dy/dx + y= 0

(d^{2}y/dx^{2})^{4} = xy

Here 1 & 2 are 4th degree and 3rd is 2nd degree

## Linear and nonlinear differential equation

A differential equation in which the dependent variable and all its derivatives are present occur in the 1st degree only and no products of dependent variables or derivatives occur known as the linear differential equation

A linear Differential equation is an equation that is not linear

## Real life application of differential equations

- The law of exponential change
- Radioactivity
- The half-life of a radioactive element
- Radioactivity, carbon dating
- Temperature Rate of change
- Electric circuit

### Application of differential equation in law of exponential change

Suppose “y” represents a quality (velocity, temperature, ….) That increases or decreases at a rate that at any time “t” is proportional to the amount present.

Suppose that “y” take the value y₀ at t = 0, there we can find “y” is a function of “t”

By solving the following initial value problem

dy/dt = ky

y = y_{0} when t = 0

*y* is positive and increasing then* k* is positive and the rate of growth is proportional to what has already been accumulated.

If *y* is positive and then decreasing* k* is negative and the rate of decay is proportional to that amount still left.

Solving the initial value problem

Integrating both sides

ln y = kt + lnC

y = eᵏᵗ × eᶜ

y = Aeᵏᵗ

To determine A we have to use the initial condition y=y₀ when t=0 putting the t=0 the above equation we get

y = y_{0} eᵏᵗ

From the above arrangement, it follows that the exponential change y = y_{0} eᵏᵗ

Is growth if k>0. and Is decay if k<0

In biology, it is often observed that the rate at which certain bacteria grow is proposition all to the number of bacterial presents at any time.

dy / dt = ky

(1/y)(dy / dt) = k

dy/y = k dt

### Radioactivity

When an atom emits some of its mass as radiation. A reminder of the atom reform to mack an atom of some new element.

This process of radiation and change is called radioactive decay and

An element whose atoms go spontaneously through this process is called radioactive.

The experiment has shown that at any given time the rate at which a radioactive element decays is approximately proportional to the number of radioactive nuclei present

Thus decay is described by the equation

dy / dt = -ky k<0

If y₀ is the number of radioactive nuclei present at time zero. Then the number of radioactive nuclei present after time “t”

Is, y = y₀e⁻ᵏᵗ k<0

### The half-life of a radioactive element

The half-life of a radioactive element is the time required for half of the radioactive nuclei present in a sample to decay

Suppose y₀ is the number of radioactive nuclei initially present in the sample and the number still present at any later time t.y is the number of nuclei present equals half the original number y₀

y₀e⁻ᵏᵗ = y_{0} / 2

e⁻ᵏᵗ = 1/2

- kt = – ln 2

t = (ln 2)/k

half-life = (ln 2)/*k*

From the derivation of half-life, it follows that the half-life is a constant that does not depend on the number of radioactive nuclei present in the sample but only on the radioactive substance

### Application of differential equations in radioactivity, carbon dating

The science of radio geology applies our knowledge of radioactivity to geology

It is known that uranium 238 undergoes radioactive decay with a half-life T = 4.55 billion years. During decay, it becomes radium 226

Similarly, C₁₄ is to radioactive carbon. Carbon in CO₂ changes to C₁₄ due to cosmic rays in the atmosphere. This Carbon dioxide is absorbed by plants. Also, radioactive carbon is accurate in animals by eating plants

In living cell rate of integration of C₁₄ exactly balanced with the rate of disinfection

So when an organisation die it use to ingest C₁₄, its C₁₄ carbon concentration decreases through the disintegration of C₁₄ present

This method used to estimate the age of organic materials and animal fossils also use to find the age of charcoal

### Application of differential equations in temperature rate of change (Newton’s law of cooling)

Under certain conditions temperature rate of change of body is proportional to the difference between the temperature of the body and the temperature of the surrounding medium

dT / dt = k(T – T_{0})

dT / (T – T_{0}) = k dt

log(T – T_{0}) = kt + C

T – T_{0} = e^{kt + C}

T = T_{0} + e^{kt + C}

We can find the death time of a person with a temperature difference.

Normal body temperature is 37°C after death it decreases with time.

The temperature changes in the atmosphere are constant then easily find out the death time

### Application of differential equations in electric circuit

In a series circuit containing a resistor and an inductor only Kirchoff’s second law state that the sum of the voltage drop across the inductor “L(di/dt)” and the voltage drop across the resistor “iR” is the same as the input voltage E(t) on the current(i) the differential equation is

L (di/dt) + iR = E(t)