Linear Differential Equation

• If the differential equation has a variable and its derivative and some other few functions are called a linear differential equation

• The standard form of a linear differential equation is represented by dy/dx + Ry = Q 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 EQUATION?

• The standard form of a 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

• The first-order differential equation is called a first-order 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 dy/dx + R1x = S

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

1.  dy/dx + 7y = Cox,

2.  dy/dx + (-5y)/x = x.e-x

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

1. dx/dy + 10x = Siny,

2.  dx/dy + 7x/y = ey. 

3. dx/dy + x/(ylogy) = 1/y.

DERIVATION FOR SOLUTION OF LINEAR DIFFERENTIAL EQUATION

The below steps are easy to understand the linear differential equation. The first-order differential equation is called a first-order linear differential equation. It is given as

          dy/dx + Rx = S

Multiply x say g(x) on both sides. The right-hand side of the derivative is given as 

     y.g(x). d/dx(y.g(x)) = y.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)]

RHS is obtained by using the products of function derivation

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

R.g(x) = go(x)

R = go(x)/g(x)

Integrate both sides with concern to x, we get

∫R.dx=∫g′(x) / g(x).dx

∫R.dx=log(g(x))

g(x)=eR.dx

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

If you substitute g(x) we get

eR.dx. bydx + R eR.dxy  = S. eR.dx

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

Integrate both sides concerning x

y.eR.dx=(S. eR.dx.dx)  

Y= e-R.dx(S. eR.dx.dx)  

The above equation is general

FORMULA OF 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 = eR.dx

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

X *( integrating factors) = {Q*(integrating factors). dy} + c, where integrating factors = er.dy

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 dry/dx + Ry = S where y is the variable, R and S are the differentials equation are either numeric constant or the function of x

STEP 2: find IF( integrating factor) where IF = eR.dx

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 the solve the linear differential functions.

HOMOGENEOUS DIFFERENTIAL EQUATIONS

• The equation that contains differentiation, set of variables, and function(x, y) is called a homogenous differential equation.

• The general form of the homogenous differential equation is as below

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

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

Homogenous differential equation

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

                dx/dy = F(x,y)

Homogenous function

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

                    where δ  is non zero constant

•  no constant term is present in the homogenous differential equation; only in the linear differential equation, and you can see the constant term

• If we remove the constant term from the linear differential equation, then the equation would turn into the homogenous differential equation

• No variable is present in the special functions like logarithm of trigonometric

HOMOGENOUS DIFFERENTIAL EQUATION EXAMPLE

Here are some example 

• dy/dx = (6x + y)/(10x – y)

• dy/dx = x(5x -12y)/y2

• dy/dx = (2x2 + 5y2)/by

• dy/dx = (3x + 9y)/(x – 2y)

• dy/dx = (11x3 + y3)/(5xy2 + 6yx2)

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

• x = δx

• y = δy

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

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

NON- HOMOGENOUS DIFFERENTIAL EQUATION

• It is much 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 y

The general solution is given as

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

where integrating factors = eR.dx

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 derivates are represented by 

dy/dx =f(x,y)= y’

 It has many applications in real-time in the newton 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

A differential equation is an equation with one or more variables and there derivatives.The greatest derivative of a differential equation is called degree and the power of greatest derivative is called the order.Differential equations can be of first order,second order and many more.In this article we learnt some ways to solve first order differential equation.