Linear Differential Equation

Introduction

A general differential equation of order n, in the dependent variable y and the independent variable x, is an equation that can be expressed in the form

a0(x)dnydxn+a1(x)dn-1ydxn-1+… ..+an-1(x)dydx+an(x)y=b(x)

The standard form of a first order linear differential equation is represented by dy/dx + Ry = S where: 

1. y is the variable
2. R and S are the differentials equation are either numeric constant or the function of x

It is one of the important forms of the differential equation, but the linear differential equation can easily solve problems.

What Is The Linear Differential Equation?

• The first order differential equation is called as a  linear differential equation
• The above example of the linear differential equation is in y we can also do in x
• The linear differential equation in x is dx/dy + R1x = S

Some examples of linear differential equations in y are given below are

1.  dy/dx + 7y = cos x
2.  dy/dx + (-5y)/x = xe-x

Some of the examples of linear differential equations in x are given below are

1. dx/dy + 10x = Siny,

Derivation For Solution Of Linear Differential Equation

The below steps are easy to understand the linear differential equation. The first order differential equation is also known as first order linear differential equation. It is given as

          dy/dx + Ry = S

Multiply both sides by a function of x, say g(x) .                                                        

g(x).dy/dx + R.g(x).y = S.g(x)  

RHS must be the derivative of the y.g(x). So choose g(x) accordingly

g(x).dy/dx + R.g(x)y = d/dx(y.g(x)]

By using the derivative formula for the product of functions, RHS is obtained in the above expression

g(x).dy/dx + R.g(x).y = g(x).dy/dx + y.g'(x)

R.g(x) = g’(x)

R = g’(x)/g(x)

Integrate both sides with concerning to x, we get

∫R.dx=∫g′(x) / g(x).dx

∫R.dx=log(g(x))

g(x)=eR.dx

This function g(x)=  eR.dxis called the Integrating Factor (I.F) 

If you substitute g(x) we get

eR.dx. dydx + R eR.dxy  = S. eR.dx

ddx(y.eR.dx)= S. eR.dx

Integrate both sides concerning x

1. eR.dx=(S.eR.dx .dx)

Y= e-R.dx(S. eR.dx.dx) + C 

The above equation is general

General Solution Formula For Linear Differential Equation

Here is the general formula to solve the linear differential equation

• The first order differential equation is called a first order linear differential equation is given as  (dy/dx) + Ry = S where R, S are the constant or the function of y

The general solution is given as

   Y * ( integrating factors) ={Q * (integrating factors).dx} + c

where integrating factors = eRdx

• If the linear differential equation dx/dy + Rx  = S then 

X *( integrating factors) =  Q* integrating factors.dy+C

where integrating factors = eRdy

Steps To Solve Linear Differential Equation

Here are three simple steps to solve linear differential equations.

STEP 1: Simply the given data in the question and write it in the form of dy/dx + Ry = S where y is the variable, R and S in the differential equation are either numeric constant or the function of x

STEP 2: find IF( integrating factor) where IF = eRdx

STEP 3: Solve the given equation and write the results as follows

Y * ( integrating factors) = {Q * (integrating factors).dx} + c

These are some of the useful steps to solve the linear differential functions.

Homogeneous Differential Equations

• The equation that contains differentiation, set of variables, and homogeneous function(x, y) is called a homogeneous differential equation.
• The general form of the homogeneous differential equation is as below

               f(x,y).dy + g(x,y).dx = 0

• The homogeneous differential equation should have the same power for the given variables (x, y)

Homogeneous differential equation

               f(x,y).dy + g(x,y).dx = 0

                dx/dy = F(x,y)

Homogeneous function

                    f (δx,δy) =  δ n f(x,y)

                    where δ  is non zero constant

•  no constant term is present in the homogeneous differential equation; it is only in the linear differential equation  you can see the constant term
• If we remove the constant term from the linear differential equation, then the equation would turn into the homogeneous differential equation
• No variable is present in the special functions like logarithm ,trigonometric

Homogeneous Differential Equation Example

Here are some example 

• dy/dx = (6x + y)/(10x – y)
• dy/dx = x(5x -12y)/y2
• dy/dx = (2×2 + 5y2)/xy
• dy/dx = (3x + 9y)/(x – 2y)
• dy/dx = (11×3 + y3)/(5xy2 + 6yx2)

You can substitute x and y in all the above examples to prove the homogeneous differential equation. 

• x = δx
• y = δy

Substitute x/y = v or x= vy when the Homogeneous differential equation is in the form of dx/dy=f(x,y) and has the homogeneous function f(x,y).

Then carry the integration part and substitute the values in the variable x,y to solve the homogeneous differential equation.

Non- Homogenous Differential Equation

• It is similar to that of a linear equation, and the order of the differential equation is not similar.
• For example
• The differential equation of the form

            (dy/dx) + Ry = S where R, S are the constant or the function of x

First Order Differential Equation

The example of a first order differential equation is given below. There only one first derivative dy/ dx is present.

For example – (dy/dx) = tan x

dy/dx = (6x + y)/(10x – 7y)

the first order derivatives are represented by 

dy/dx =f(x,y)= y’

 It has many applications in real-time in Newton’s law of cooling and electric circuits.

Second Order Differential Equation

Here is an example of a second-order differential equation. The equation has second-order derivative dy/ dx is present. It is represented as 

=d/dx(dy/dx)

=d2y/dx2

=f”(x) = y”

For example – (5d2y/dt2) + (8d2x/dt2) = 12x

Conclusion

In the topic linear differential equation we have learnt the representation of linear equations along with examples. We also saw how to solve first order and second order linear equations using the General Formula. We learnt about homogeneous and non homogeneous equations.