JEE Main 2025: Differential Equations Notes

The  Differential Equations chapter covers the study of mathematical equations that involve derivatives and their solutions, used to model a wide range of real-world phenomena and scientific principles. This is one of the important chapters for JEE Main Exams, and you can expect a minimum of 2 questions from this chapter. Look at the number of questions asked from the Differential Equations chapter in the past five years of the JEE Main exam.

Take the chance to elevate your preparation with our Differential Equations Notes. It covers the JEE Main Maths syllabus, which you can download for FREE. 

Top Concepts to Study in Differential Equations for JEE Main 2025

Here you will review the main topics in Differential Equations. These crucial topics can truly improve your preparation. Good at them will give you a big advantage in the upcoming test. So, let's discuss the essential topics you need to consider for success in JEE Main 2025.

• Variable Separable Equations

• Homogeneous Differential Equations

• Linear Differential Equations

• Exact Differential Equations

• Integrating Factor Method

• Second-Order Differential Equations

• Reduction of Order

• Variation of Parameters

• Applications in Physics and Engineering

• Systems of Differential Equations

• Maths Notes: Chapter-wise links 2025

The Significance of Differential Equations Notes for JEE Main 2025

Amidst the rigorous preparation for JEE Main 2025, the "Differential Equations" chapter emerges as a cornerstone of mathematical understanding, with profound implications extending beyond the exam. Comprehensive notes on this chapter not only bolster your exam performance but also lay the groundwork for analytical thinking, problem-solving skills, and real-world applications.

• Fundamental Modelling Tool: Notes elucidate how differential equations serve as a powerful tool for modelling real-world phenomena, from physics and engineering to biology and economics.

• Predictive Precision: By mastering differential equations, you gain the ability to predict and understand the behaviour of dynamic systems, enriching your scientific insight.

• Real-life Problem Solving: Notes enable you to tackle intricate problems that mirror real-world challenges, from growth patterns to harmonic oscillations, fostering practical problem-solving skills.

• Bridge Between Mathematics and Science: Differential equations bridge mathematics with physics, engineering, and various scientific disciplines, cultivating interdisciplinary thinking.

• Engineering and Physics Applications: These notes empower you to solve problems related to motion, electricity, circuits, mechanical systems, and more—skills invaluable in engineering and physics.

• Holistic Mathematical Understanding: By grasping the core concepts of differential equations, you deepen your mathematical acumen and expand your analytical toolkit.

• Higher-order Thinking: Differential equations challenge you to think beyond basic algebra, honing your ability to navigate complex mathematical scenarios.

• Preparation for Further Studies: The concepts covered prepare you for more advanced mathematical topics in university studies and research.

• Analysing Change: Differential equations focus on how quantities change, nurturing your understanding of dynamic processes and transformations.

• Academic and Career Relevance: Proficiency in differential equations is sought after in fields such as engineering, physics, economics, and environmental sciences.

• Long-term Intellectual Growth: While integral for JEE Main success, mastering differential equations serves as a foundation for continuous intellectual growth beyond the exam.

It's evident that studying the Differential Equations chapter before JEE Main 2025 is crucial. You can grasp the concepts effortlessly by downloading the Differential Equations Notes and conducting a fast review.

Formula-Memory Connection: Strengthening Knowledge with Revision Notes

Equations are more than just symbols –They're your secret code to solve problems in Maths. When you understand formulas, you can easily handle tricky ideas and feel more sure of yourself. Use the equation with JEE Main Maths Revision notes to improve studying. Check out our JEE Main Maths formula pages and begin your journey to prepare for JEE Main now!

Here are the key Differential Equations formulas and principles relevant for JEE Main:

 1. Basic Concepts

• A differential equation involves derivatives of a function. For example, $\dfrac{dy}{dx} = f(x)$.

• Order: The highest derivative in the equation.

• Degree: The power of the highest derivative, provided it is a polynomial in derivatives.

 2. General Solution and Particular Solution

• General Solution: A solution containing arbitrary constants.

• Particular Solution: A solution obtained by assigning specific values to the arbitrary constants.

3. Types of Differential Equations

 First-Order Differential Equations

1. Separable Differential Equations:  

• Form: $\dfrac{dy}{dx} = g(x)h(y)$

• Solution: Separate variables and integrate:

• \[\int \dfrac{1}{h(y)} \, dy = \int g(x) \, dx\]

2. Homogeneous Differential Equations:  

• Form: $\dfrac{dy}{dx} = f\left(\dfrac{y}{x}\right)$

• Solution: Substitute $ y = vx $ (where $ v = \dfrac{y}{x} $), then solve by separation of variables.

3. Linear Differential Equations:  

• Form: $\dfrac{dy}{dx} + P(x)y = Q(x)$

• Solution: Use the integrating factor (IF), given by:

• \[\text{IF} = e^{\int P(x) \, dx}\]

• Then, the solution is:

• \[y \cdot \text{IF} = \int Q(x) \cdot \text{IF} \, dx + C\]

4. Exact Differential Equations:  

• Form: $ M(x, y) \, dx + N(x, y) \, dy = 0 $

• Condition: $ \dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x} $

• Solution: Find a function $ \psi(x, y) $ such that $ d\psi = M \, dx + N \, dy $.

 4. Second-Order Differential Equations with Constant Coefficients

• Form: $ a \dfrac{d^2y}{dx^2} + b \dfrac{dy}{dx} + c y = 0 $

• Auxiliary Equation: $ ar^2 + br + c = 0 $

• If roots $ r_1 $ and $ r_2 $ are real and distinct:

• \[y = C_1 e^{r_1 x} + C_2 e^{r_2 x}\]

• If roots $ r_1 = r_2 = r $ (real and equal):

• \[y = (C_1 + C_2 x)e^{r x}\]

• If roots are complex $ r = \alpha \pm \beta i $:

• \[y = e^{\alpha x} (C_1 \cos \beta x + C_2 \sin \beta x)\]

 5. Formation of Differential Equations

• To form a differential equation from a family of curves, differentiate the given equation as many times as there are arbitrary constants, and then eliminate these constants.

 6. Solution of Differential Equations by Substitution

• Clairaut’s Form: For equations of the form $ y = x \dfrac{dy}{dx} + f\left(\dfrac{dy}{dx}\right) $, find solutions by differentiating or substituting accordingly.

 7. Applications of Differential Equations

• Growth and Decay: $\dfrac{dy}{dt} = ky$, where $k$ is the growth/decay constant.

• Solution: $ y = y_0 e^{kt} $

• Newton’s Law of Cooling: $\dfrac{dT}{dt} = -k(T - T_{\text{env}})$, where $T_{\text{env}}$ is the environment temperature.

Explore Further: Supplementary Resources for JEE Main 2025 Differential Equations

Looking to enhance your JEE Main 2025 Differential Equations preparation? Explore these Important materials and Use these tools to improve comprehension and test-taking skills, boosting your JEE Main 2025 performance!

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