# Differential Equations Applications

Differential Equation applications have significance in both academic and real life. An equation denotes the relation between two quantity or two functions or two variables or set of variables or between two functions. Differential equation denotes the relationship between a function and its derivatives, with some set of formulas. There are many examples, which signifies the use of these equations.

The functions are the one which denotes some sort of operation performed and the rate of change during the performance is the derivative of that operation, and the relation between them is the differential equation. These equations are represented in the form of order of the degree, such as first order, second order, etc. Its applications are common to find in the field of engineering, physics etc.

In this article, we will learn about various applications in real life and in mathematics along with its definition and its types.

## Differential Equations

In terms of mathematics, we say that the differential equation is the relationship that involves the derivative of a function or a dependent variable with respect to an independent variable. It is represented as;

\(\frac{d(y)}{d(x)}\) = f(x) = y’Or

y’=\(\frac{d(y)}{d(t)}\)

Or

f(x,y) = \(\frac{d(y)}{d(x)}\) = \(\frac{d(y)}{d(t)}\) = y’

Or

x_{1}\(\frac{d(y)}{d(x_{1})}\) + x_{2} \(\frac{d(y)}{d(x_{2})}\) = y

Where x is the independent variable

And y is the dependent variable, as its function is dependent on the values of x.

Y’ denotes one derivative. Similarly, y’’, y’’’, …, so on, denoted number of derivatives for all values of x.

There are many applications of differential equations in mathematics, based on these formulas.

## Types of differential equations

Basically, there are two types of differential equations;

**Ordinary Differential Equation(ODE)**

Ordinary differential equation involves a relation between one real variable which is independent say x and one dependent variable say y and sum of derivatives y’, y’’, y’’’… with respect to the value of x.

f(x) = y = \(\frac{d(y)}{d(x)}\)

The highest derivative which occurs in the equation is the order of ordinary differential equation. ODE for nth order can be written as;

F(x,y,y’,….,y^{n}) = 0

Ordinary differential equations applications in real life are used to calculate the movement or flow of electricity, motion of an object to and fro like a pendulum, to explain thermodynamics concepts. Also, in medical terms, they are used to check the growth of diseases in graphical representation.

For example, according to the Newton law of cooling, the change in temperature is directly proportional to the difference between the temperature of the hot object or body and temperature of the atmosphere. Therefore, in terms of the differential equation we can represent it as;

\(\frac{dT}{dt}\) \(\propto\) (T – T_{a})

Or

\(\frac{dT}{dt}\) = k (T – T_{a})

Where k is the proportional constant and T is the temperature of the object and T_{a }is the temperature of the atmosphere

**Partial Differential Equation(PDE)**

In the partial differential equation, unlike ordinary differential equation, there is more than one independent variable.

For example:

- \(\frac{dz}{dx}\) + \(\frac{dz}{dy}\) = 2z is a partial differential equations of one order.
- \(\frac{d^2u}{dx^2}\) + \(\frac{d^2u}{dy^2}\) + 2x + 2y – z is a partial differential equation of second order.

## Applications of Differential Equations

We can describe the differential equations applications in real life in terms of:

**Exponential Growth**

For exponential growth, we use the formula;

**G(t)= G _{0} e^{kt}**

Let G_{0 }is positive and k is constant, then

G(t) increases with time

G_{0 }is the value when t=0

G is the exponential growth model.

**Exponential reduction or decay**

**R(t) = R _{0} e^{-kt}**

When R_{0 }is positive and k is constant, R(t) is decreasing with time,

R is the exponential reduction model

Newton’s law of cooling, Newton’s law of fall of an object, Circuit theory or Resistance and Inductor, RL circuit are also some of the applications of differential equations.

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