Fourth Order Differential Equation

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An equation that contains, ordinary or partial derivative then it is said to be differential equation.general form of an differential equation can be given as "`f(x)= dx/dt` " . There are many orders of differential equation namely first order, second order , third order , fourth order and so on. Now in this chapter we are going to discuss about the fourth order differential equation. In order to know the fourth order differential equation we need to know about the other lower order equation. Now lets discuss about the orders of differential equation.

Method to Solve Fourth Order Differential Equation:

First order differential equation:

The general form of first order differential equation is given by

` f(a,b)=(db)/(da)`

Lets solve an example problem to better understand the first order differential equation

Example 1:

Calculate the first order derivative of sin(2x)?

Solution:

Given:

` f(x)=sin(2x)`

To find:

`d/dx(sin(2x))`

Now to calculate the first order follow the steps below

Step 1:

First we need to differentiate the ...
... sin value

i.e. differentiation of sin x = cos x

Now the above problem become

`cos2x(d/dx(2x))`

Step 2:

Now after calculating the sin value then we need to calculate the inner x value,

so on differentiating the 2x we get,

(i.e.` d/dx(x^2)` = `x^(2-1)` =`x` )

By following the above step we get

` d/dx(2x) ` = `2`

Thus combing the above two answer we get

`d/dx(sin2x)` =`2cos(2x) `

Thus the first order derivative is known.

Second order differential equation:

The general form of second order differential equation is given by

`f(x,y) =( (d^2y)/dx^2)+((dy/dx))`

Now lets solve an problem to better understand the second order derivative.

Example 2:

Calculate the second order derivative of `x^2+x` ?

Solution:

Given:

`f(x)=x^2+x`

To find:

` d^2/dx^2(f(x))` = `d^2/dx^2(x^2+x)`

Step 1:

To find the second order derivative we need to first find the first order derivative

i.e. `d/dx(d/dx(x^2+x))`

`d/dx(x^2+x)` =`x^(2-1)+x^(1-1)`

`d/dx(x^2+x)` =`x+1`

Step 2:

Now first order derivative is known ,then find the second order derivative for obtained answer,

i.e. once again differentiate the `(x+1)`

` d/dx(x+1)` =`x^(1-1)+0`

`d/dx(x+1)=1`

Thus the second order derivative is found.

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Some Example to Solve Fourth Order Differential Equation:

Third order differential equation:

The general form of third order differential equation is given by

` f(x,y)=(((d^3y)/dx^3)+((d^2y)/dx^2)+((dy)/dx))`

Now lets solve some example to know it better

Example 3:

Calculate the third order derivative of `(x^3)+(x^2)+(x)` ?

Solution:

Given:

` f(x)=x^3+x^2+x`

To find:

` d^3/dx^3(f(x))=d^3/dx^3(x^3+x^2+x)`

Step 1:

First find the first order derivative,

i.e. `d^2/dx^2(d/dx(f(x)))=d^2/dx^2(d/dx(x^3+x^2+x))`
= `x^(3-1)+x^(2-1)+x^(1-1)`
`d/dx` = `x^2+x+1`

Step 2:

Now find the second order derivative,

`d/dx(d/dx(x^2+x+1))=d/dx(x^(2-1)+x^(1-1)+0)`

` = d/dx(x+1)`

Step 3:

Now the third order can be found by again differentiating the second order value,

`d/dx(x+1)=x^(1-1)+0`

` =1`

` d^3/dx^3(f(x)=1`

Fourth order differential equation:

The general form of fourth order differential equation is given by,\

` f(x,y)=((d^4y)/dx^4)+((d^3y)/dx^3)+((d^2y)/dx^2)+((dy)/dx)`

Now lets solve some problem for better understanding,

Example 4:

Calculate the fourth order derivative of `x^4+x^3+x^2` ?

Solution:

Given:

` f(x)=x^4+x^3+x^2`

Step 1:

First order derivative,

`d/dx(f(x))=d/dx(x^4+x^3+x^2)`

`= x^(4-1)+x^(3-1)+x^(2-1)`

` d/dx(f(x))=x^3+x^2+x`

Step 2:

Find the second order derivative,

` d^2/dx(d/dxf(x))=d^2/dx(x^4+x^3+x^2)`

` =x^(3-1)+x^(2-1)+x^(1-1)`

`= x^2+x+1`

Step 3:

Find the third order derivative,

` d/dx(d/dx(x^2+x+1)=d/dx(x^(2-1)+x^(1-1)+0)`

` =d/dx(x+1)`

Step 4:

Now the final step of finding the fourth order derivative,

`d/dx(x+1)=x^(1-1)+0`

`=1`

Thus the fourth order derivative is given by,

` d^4/dx^4(f(x))=1`

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