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.
First, arrange the given 1st order differential equation in the right order (see below)
dy/dx + A(y)= B(x)
Pick out the integrating factor, as in, IF= e ∫A(y)dx
Multiply given equation with IF. (Use the product rule to solve the equation)
Integrate both sides concerning x.
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
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.
A differential has an infinite number of solutions which stems from the fact that a function can have an infinite number of antiderivatives.
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.
FAQs on First Order Differential Equation
1. What is a first order differential equation, and how is it represented in the CBSE Class 12 Maths syllabus?
A first order differential equation is an equation involving the derivative of a function with respect to one variable, where the highest derivative present is of the first order (dy/dx). In the CBSE Class 12 Maths syllabus, these equations are represented as dy/dx = f(x, y) and are solved using various methods such as separation of variables, integrating factor, and homogeneous equations.
2. How do you solve a first order linear differential equation using the integrating factor method?
To solve a first order linear differential equation of the form dy/dx + P(x)y = Q(x), follow these steps:
- Find the integrating factor (IF): e∫P(x)dx
- Multiply both sides of the equation by the IF.
- The left side becomes the derivative of (IF × y).
- Integrate both sides with respect to x.
- Solve for y to obtain the general solution.
3. What are the main applications of first order differential equations in real-life scenarios?
First order differential equations play a crucial role in modelling various real-life phenomena. Examples include:
- Physics: Describing velocity and acceleration under Newton’s laws.
- Biology: Modelling population growth and decay.
- Economics: Studying interest rates and investment growth.
- Chemistry: Describing rates of reactions.
4. Why is it important to learn different methods of solving first order differential equations for board exams?
Board exams often test understanding of various methods such as separation of variables, homogeneous equations, and integrating factors, each suited to specific forms of first order differential equations. Mastery of these techniques enables you to tackle a wider range of questions accurately and demonstrates conceptual depth, which is essential for full marks in CBSE 2025–26 exams.
5. What are common mistakes students make while solving first order differential equations, and how can they be avoided?
Frequent errors include incorrect calculation of the integrating factor, forgetting the constant of integration after integrating, and misidentifying the type of equation. To avoid these, always:
- Check the form of the equation before choosing a method.
- Carefully compute the integrating factor or separation steps.
- Add the constant of integration after every indefinite integration.
6. How can first order differential equations appear in CBSE board exams, and what is their typical marks weightage?
Questions on first order differential equations in CBSE board exams usually appear as 3-mark or 5-mark problems, requiring students to solve an equation using the correct method and present all steps clearly. Sometimes, HOTS or application-based questions related to real-world models may also be asked.
7. What is the difference between homogeneous and non-homogeneous first order differential equations, and how does the solution approach vary?
A homogeneous first order differential equation has terms with the same degree (e.g., dy/dx = f(y/x)), and is solved by substituting y = vx. In contrast, non-homogeneous equations contain additional functions of x or y. The approach varies—homogeneous ones use substitution, while non-homogeneous often require integrating factors or other specific techniques.
8. Can you explain why separation of variables is an effective technique for certain first order differential equations?
Separation of variables is effective when the equation can be written so that all terms involving y are on one side and all terms involving x are on the other. This allows for direct integration of both sides, leading to a straightforward solution. It's particularly useful for equations like (dy/dx) = g(x)h(y).
