Linear Differential Equations

The linear polynomial equation, which consists of derivatives of several variables is known as a linear differential equation.

The solution of a differential equation is the term that satisfies it. It can also be the case where there are no solutions or maybe infinite solutions to the differential equations.

What are Differential Equations? 

Let us understand this with an example -

dx/dt denotes the instantaneous rate of change.

We will now make an assumption about the rate of changes or variations with first powers of x

dx/dt = -km

k>0 is a proportionality constant.

dx/x = -kdt……………(1)

Integrating it we get

Log = -kt + c

Where c is an arbitrary constant.

x = e^-kt+c

x = Ae^-kt

You must have seen this term in chemistry.

When we write dy/dx then y is a dependent variable and x is the independent variable. We can clearly see that the variations in y are dependent on x.

The term dy/dx denotes the rate of change of y with respect to x.

Now, we can say that an equation that involves dependent variables, independent variables, and derivatives of the dependent variable is called differential equations. It relates one or more functions and their derivatives.

Order And Degree In Differential Equations

  • The degree is the highest power of derivative.

  • Order is the highest numbered derivative

In this article, we are going to see about first-order linear differential equations.

Types of Differential Equations

Based on the approach to the solution we can classify these differential equations in several types such as


2.Linear/Non-linear, and Homogeneous/heterogeneous.

First-order Linear Differential Equations

When there is only the term dy/dx used in the equation then it is termed as the first-order linear differential equation. The function y and its derivatives that are present in the equation are of up to first degree only. There are no products of y or any of its derivatives present and also the transcendental functions of y also do not occur.

Integrating Factor

A multiplying factor which will convert an inexact differential equation into an exact one is called the integrating factor. 

∫p(x) dx = log u(x)

∴ u(x) = e^∫p(x)dx

This u(x) is also called the integrating factor.

The General Form Of Linear Differential Equations

1. dy/dx + py = q

p,q are continuous  functions of x or constants

I.F = e^∫pdx

solution of this form is 

y(I.F) = ∫(I.F)Q dx

2. dx/dy + px = q

p,q are continuous  functions of y or constants 

I.F = e^∫pdy

solution of this form is 

x(Ι.F) = ∫(I.F)Q dy

Example : dy/dx + y/x = logx/x

1.First, we need to find whether it is a linear differential equation.

2.Find the type of equation it is. 

we can clearly see that it is of the type

dy/dx + py = q

and p,q are functions of x 

3.Identify p,q 

∴ p = 1/x and q = logx/x 

I(x) = e^∫p(x)dx

= e^∫1/x dx 

= e^ log(x)


4.solution of this form is 

y(I.F) = ∫(I.F)Q dx

y(x) = ∫logx/x . x + c

∴ y(x) = ∫log(x) + c

∴ y(x) = x log x - x + c

Method for Finding a General Solution of the General First-order Linear Differential Equations.

By multiplying the integrating factor on both sides we get

[p(x)e^∫p(x)dxy - q(x)e^∫p(x)dx]dx + e^∫p(x)dxdy = 0

rewriting above equation in this form

e^∫p(x)dx dy + p(x)e^∫p(x)dx . ydx = q(x)e^∫p(x)dx. dx

∴ e^∫p(x)dx. y = ∫ e^∫p(x)dx .q(x)dx + c

FAQ (Frequently Asked Questions)

1. Do we Need to Add the Constant of Integration in the Integrating Factor?

Answer: No, we do not add the constant of integration after we finish with making integrating factors. The constant of integration is considered to be zero.

2.How to Solve First-order Linear Differential Equations?

Step 1: First of all, we need to write the given equation in standard form ie

y' + p(x)y = q(x)

Step 2: Identify the functions p(x) and q(x)

Step 3: Next, we need to find the integrating factor that is 

I(x) = e^∫p(x)dx

Step 4: Solve the question by using the above solution method that is 

y(I.F) = ∫(I.F)Q dx

3.How do you Find Integrating Factor?

Answer: Steps to find the integrating factor are-

1. Assume, we have the equation

dy/dx + p(x)y = q(x)

2. Now we rewrite this equation

[p(x)y - q(x)]dx + dy = 0

3.Then multiply it with a term u(x)

[u(x)p(x)y - u(x)q(x)] dx + u(x)dy = 0

4. We know by definition of integrating factor 

A term is called I.F if it makes the differential equation exact.

By theorem of exactness we get ,

δ/δy [u(x)p(x)y - u(x)q(x)] = δ/δx u(x)

taking these derivatives [ we must remember p(x),q(x) and u(x) are only the functions of x ]

Taking these derivatives we obtain

u(x). p(x) = d/dx. u(x)

We can write this as

p(x) dx = du(x)/u(x)

Integrating both sides we get

∫p(x) dx = log u(x)

∴ u(x) = e^∫p(x)dx

This u(x) is also called the integrating factor.