### Notes from Toppers

# Differential Equations [JEE Advanced]

**1. First Order Differential Equations**

**Solving linear first-order differential equations using an integrating factor:**- Learn to identify linear first-order differential equations and apply the integrating factor method.
- Practice solving problems involving integrating factors. Refer to NCERT Class 12, Chapter 9, Section 9.5 for the theoretical background.
**Solving first-order exact differential equations:**- Understand the concept of exact differential equations and learn to identify them.
- Practice solving exact differential equations by integrating both sides. Refer to NCERT Class 12, Chapter 9, Section 9.6 for the theoretical background.
**Solving linear differential equations using separation of variables:**- Learn the method of separation of variables and apply it to solve linear differential equations.
- Practice solving problems involving separation of variables. Refer to NCERT Class 12, Chapter 9, Section 9.7 for the theoretical background.

**2. Higher Order Linear Differential Equations**

**Solving linear homogeneous differential equations with constant coefficients:**- Learn the auxiliary equation method and apply it to solve homogeneous differential equations with constant coefficients.
- Practice solving problems involving homogeneous differential equations. Refer to NCERT Class 12, Chapter 9, Sections 9.8 and 9.9 for the theoretical background.
**Solving linear non-homogeneous differential equations with constant coefficients using the method of undetermined coefficients or variation of parameters:**- Learn the methods of undetermined coefficients and variation of parameters, and apply them to solve non-homogeneous differential equations with constant coefficients.
- Practice solving problems involving non-homogeneous differential equations. Refer to NCERT Class 12, Chapter 9, Section 9.10 for the theoretical background.

**3. Systems of Differential Equations**

**Solving systems of linear differential equations using matrix methods:**- Learn to solve systems of linear differential equations using matrices and eigenvalues.
- Practice solving systems of linear differential equations using matrices. Refer to NCERT Class 12, Chapter 9, Section 9.11 for the theoretical background.
**Solving systems of nonlinear differential equations using graphical methods or numerical methods:**- Understand graphical methods and numerical methods for solving systems of nonlinear differential equations.
- Practice solving systems of nonlinear differential equations using graphical or numerical methods. Refer to relevant resources for the theoretical background.

**4. Laplace Transforms**

**Definition and properties of Laplace transforms:**- Learn the definition and basic properties of Laplace transforms.
- Understand the concept of the Laplace transform of a function. Refer to NCERT Class 12, Chapter 10, Section 10.1 for the theoretical background.

**Solving differential equations using Laplace transforms:**- Apply Laplace transforms to solve differential equations with initial conditions.
- Practice solving differential equations using Laplace transforms. Refer to NCERT Class 12, Chapter 10, Section 10.2 for the theoretical background.
**Inverse Laplace transforms and their applications:**- Learn the concept of the inverse Laplace transform and methods to find it.
- Apply the inverse Laplace transform to solve problems involving differential equations. Refer to NCERT Class 12, Chapter 10, Section 10.3 for the theoretical background.

**5. Applications of Differential Equations**

**Modeling and solving problems in physics, engineering, and other fields using differential equations:**- Learn how differential equations are used to model real-world problems.
- Practice solving application-based problems involving differential equations from various fields. Refer to relevant textbooks and resources for specific application areas.

Remember to consistently practice solving problems, revise concepts regularly, and refer to additional study materials and resources recommended by your teachers or mentors to enhance your understanding of differential equations.