First Order Differential Equation

The root of the first order differential equation lies in the term derivative. A clear understanding of derivatives can make the process of learning differential equations easier and digestible. In mathematical terms, a derivative is a tool to measure a rate of change of values in a function at a particular point in the function.  It can also be understood as a slope, which signifies the ratio of the rate of deviation in the value of a particular function based on an independent variable. Now, it is a good time to deal with the question: what is a differential equation? 

What is a Differential Equation?

A differential equation represents the change in the derivative of a dependent variable ‘X’ concerning an independent variable ‘Y’. A differential equation may contain more than one variable.  

A differential equation is represented as 

dy/dx +f(y)= Q(x)

General Solution of 1st Order Differential Equation

The general solution of the first order differential equation provides the relation between both the dependent and the independent variable devoid of any form of derivative. Also, the relation arrived at, will inadvertently satisfy the equation at hand.

We will look into the details and talk about the different approaches to solve the first order differential equation in the later stage. 

First-Order Differential Equation

There are different orders of differential equations, but we will focus mainly on the first-order form. On a side note, it will come in handy if you know what is an order of a differential equation?

The order of a differential equation is always the highest order of derivatives. The order of the differential is the highest number attached to the 'd' which is in the numerator (for pictorial 

understanding).

A first order differential equation indicates that such equations will be dealing with the first order of the derivative.

Again for pictorial understanding, in the first order ordinary differential equation, the highest power of 'd’ in the numerator is 1.  

A first-order differential equation is one of the five different types of DE, each of them are mentioned below.

  • Separable equation

  • Integrable equation

  • Exact equations 

  • Homogeneous equation

  • First order linear differential equation

Next, we will look into first order linear differential equations. It is one of the basic elements of DE. A proper understanding of first order linear differential equations can make the process of learning DE smooth.

First Order Linear Differential Equation

Before we head any further, it is important to familiarise yourself with the most important form of DE that is First order linear differential equation. It is the simplest form first order DE, the general solution of which can be easy to find. Thus, it is advised that students do not memorize the formulae. Next, we will look into a couple of methods to solve the 1st order differential equation.

Example - dy/dx+ R(t)y= s(t)

How to Solve 1st Order Differential Equations?

Solving 1st order differential equations can be tedious but the two approaches mentioned below can ease things out. 

Method of Variation

Methods of variation of parameters deals with solving the homogeneous equation on the left-hand side to arrive at a general solution. This is done by sustaining the RHS with

zero and solving the equation accordingly.  We have mentioned steps for solving 1st order differential equations.

        y’+3y=g(t)

Step -1: Solving the homogeneous equation by substituting g(t)=0

The general solution of a first order differential equation arrived by solving the homogeneous equation, it will include a constant of integration, say K. The constant part can be substituted with an unknown

function K(x). 

Step-2: The next step is to substitute the function K(x) into the nonhomogeneous differential equation. After the substitution, the uncertain function K(x) is determined to certainty.

Integrating Factor

For solving 1st order differential equations using integrating methods you have to adhere to the following steps.

  1. First, arrange the given 1st order differential equation in the right order (see below)

dy/dx + A(y)= B(x)

  1. Pick out the integrating factor, as in, IF= e ∫A(y)dx 

  2. Multiply given equation with IF. (Use the product rule to solve the equation)

  3. Integrate both sides concerning x. 

  4. Finally, divide the equation with the IF to arrive at the final value of ‘y’.

Both methods provide the same solution of first order differential equation.

Solved First Order Differential Example

Here, we will see the first order differential equation example, so that you can have a clear idea of how to solve the first order differential equation. The below mentioned first order differential example deals with exponential function. 

Question 1 : Solve the equation y′−y−xex = 0

Solution : Given, y′−y−xex = 0

First, you have to rewrite the provided equation in the form given below.,

y′−y = xex

Using the integrating factor, it becomes;

K(x)=e∫(−1)dx=e−∫dx=e−x

Now you have proceeded in the normal using the last few steps mentioned in the integrating factor method. There are various first order differential equation example on the web, which we would recommend you to have a look at.

Did You Know

  1. Have you ever thought about the number of solutions for a differential equation? Only those who have a core understanding of the concept will know the correct answer. 

  2. A differential has an infinite number of solutions which stems from the fact that a function can have an infinite number of antiderivatives.

  3. You can carve out DE formulae by knowing the antiderivative of a function. The formula can help you to draw the graph. This concept is used in physics for studying the motion of an oscillating object.

FAQ (Frequently Asked Questions)

1. Is There Any Real-Life Application of the First Order Differential Equation? 

It is quite unfortunate that as kids, most of us learn first order ordinary differential equations without asking about its application. Newton's second law of motion f=ma can be turned into an apt first order differential equation example. 

There are various mind-bending applications of differential equations in real life, which we are completely unaware of. In broader terms, we can see the application of first order differential equation in various fields like biology, economics, chemistry, physics, and engineering. 

They can be used to determine the population growth, return on investment, modelling behavior of moving objects amongst the various other applications.

2. Is the Differential Equation Easy?

Many students skip the differential equation part altogether citing its complexity and difficulty, But in reality, if you keenly follow the lectures and develop an interest in the topic at hand, it will cease to be tough. 

The key to learning differential equations is to immerse yourself into every known sphere of the topic. This is where the interest comes into play. 

Developing interest will inspire you to research more and more. Just the sheer thirst of knowledge for the topic beyond what is taught in class will make things easier. 

So, next time when you are in the mood to skip the topic, think about how interesting it can be, and what you are missing out on.