What are Differential Equations? Easily Explained with Examples

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Differential Equations

Before starting our main topic of Differential Equations, first, we would know about the equation. We would understand that what is an equation and how can we represent it.

See Also: What is a derivative? Everything About it in One Post!

Equations

It is a mathematical statement in which two expressions are equal. For example:

  • 2x+3=5             (linear equation with one variable x)
  • 2x+3y=5           (linear equation with two variables x and y)
  • 2x+3y+4z=4     (linear equation with three variables x,y, and z)
  • x^2+3x=5         (quadratic equation with one variable)
  • x^2+y^2=5       (quadratic equation with two variables x and y)
  • sinx+cosy=1    (trigonometry equation with two variables x and y)

And so on…

Now, we will proceed towards the differential equation and its types:

Differential Equations

It is a mathematical expression that contains the derivatives of independent variables with respect to the dependent variable.

xdy/dx+y=0

xd^2 y/dx^2+y=0

xd^2y/dx^2+ydy/dx=0

Here, y is a dependent variable and x is an independent variable.

Differential Equations
Differential Equations

For example: A man moves in 10m in 8 seconds then its position is dependent while the time he takes to travel 10m distance is independent.

Similarly, with the differential equation, a differential equation must contain derivatives of the dependent variable with respect to an independent variable.

For example

y=x^2

Here, the value of y depends on the value of x. Lets put some values of x and then correspondingly, find out the values of y.

  1. When x=1                        then                    y=1
  2. x=2                                 then                    y=4
  3. x=3                                 then                    y=9
  4. x=9                                 then                    y=81
  5. x= 10                              then                    y=100

Hence, we can see that the value of y depends on the value of x. So, we can say that y is a dependent variable, and x is an independent variable.

See Also: What is a Partial Derivative? Scope and Application

Types Of Differential Equation

Here you will find different types of Differential Equations

Ordinary Differential Equation

The differential equation in which the dependent variable depends only on one independent variable.

Let say-

y=f(x)

Where y is a function of x which means y is the dependent variable and x is the independent variable.

Then the equation :

dy/dx=0

Or

                           df(x)/dx=0             …………(1)

Then the equation (1) is known as an ordinary equation.

Let’s take some examples of differential equations:

Above first 5 equations ( from 1 to 5) are known as ordinary differential equation while the differential equation no. 6 is not an ordinary differential equation. This 6th equation is known as the partial differential equation, which is the other type of differential equation.

Partial Differential Equation

The differential equation in which the dependent variable depends on more than one independent variable. Then the differential equation is known as the Partial Differential equation.

Let’s say-

y=f(x,t)

From the above equation, we can say that “y” is a function of x and t. Which means y is a dependent variable that depends on the independent variable x and t.

Then –

∂^2 y/∂x^2+∂^2 y/∂t^2=0             ……….(2)

Then the above equation (2) is known as the partial differential equation. And the symbol of this partial differential equation is “”. This type of differential equation is also known as the Laplace Equation.

Note: Partial differential equation is in higher studies not in the 12th standard

See Also: How To Find Derivative Of A Graph | Easiest Method

Notation for Differential Equation

We generally use some notations for a differential equation which are as follow:

dy/dx=y^’

d^2 y/dx^2=y”

d^3 y/dx^3=y”’

.

.

.

.

d^n y/dx^n=y^n

Order of a Differential Equation

The highest order derivative involved in differential equations is known as the order of that differential equation.

  • xdy/dx+y=0………….(it has dy/dx which is the higher-order derivative in this equation)
  • xdy/dx+sin(dy/dx)=0…..(it has dy/dx which is the higher-order derivative in this equation)
  • xdy/dx+sin(d^2 y/dx^2)=0 (it has d^2 y/dx^2 which is the higher-order derivative in this equation)
  • d^2 y/dx^2+ydy/dx=0…(it has d^2 y/dx^2 which is the higher-order derivative in this equation)

Now, here are the naming of these orders:

  • First order derivative = dy/dx
  • Second Order derivative= d^2 y/dx^2
  • Third Order Derivative= d^3 y/dx^3

  • nth Order Derivative= d^n y/dx^n

Hence, we call equations 1st and 2nd as first-order equations while the equations 3rd and 4th as the second-order equations.

Degree of Differential Equation

We call The highest power of the higher-order derivative of a differential equation as the degree of the differential equation.

  • x(dy/dx)^2+y=0…………………..(it has dy/dx which is the higher-order derivative in this equation and Also has the degree 2)
  •  xd^2 y/dx^2+sin(dy/dx)^4=0…………….(it has d^2 y/dx^2 which is the higher-order derivative in this equation and has degree 1)
  • xdy/dx+sin(d^2 y/dx^2 )^3=0 …………… (it has d^2 y/dx^2 which is the higher-order derivative in this equation and degree 3)
  • d^2 y/dx^2+y(dy/dx)^5=0…………………(it has d^2 y/dx^2 which is the higher-order derivative in this equation and has degree 1)

Hence,

  • The equation 1 is the first-order second-degree differential equation.
  • Equation 2 is the second-order first-degree differential equation.
  • The equation 3 is the second-order third-degree differential equation.
  • Equation 4 is the second-order first-degree differential equation.

Also Refer: How To Find Derivative Of A Function? [Simple Ways]

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