Differential equations are equations that involve one or more derivatives of an unknown function with respect to one or more independent variables. The Class 12 Mathematics syllabus covers the following topics related to differential equations:
Definition of a Differential Equation: This topic covers the definition of a differential equation, its order, and degree.
General and Particular Solutions of a Differential Equation: This topic covers the concept of general and particular solutions of a differential equation.
Formation of Differential Equations: This topic covers the methods for forming differential equations from given functions and the concept of parametric form.
Methods of Solving First-Order Differential Equations: This topic covers the methods for solving first-order differential equations, such as separation of variables, homogeneous equations, and exact equations.
Linear Differential Equations: This topic covers the methods for solving linear differential equations, such as the method of integrating factors.
Applications of Differential Equations: This topic covers the applications of differential equations in various fields, such as physics, chemistry, engineering, and biology.
Differential equations are important because they provide a powerful tool for modeling various physical, chemical, and biological phenomena. The concept of general and particular solutions is used in solving initial value problems, which are problems that involve finding the solution of a differential equation that satisfies certain initial conditions. The formation of differential equations is used in modeling real-world problems and in the design of control systems. The methods for solving first-order and linear differential equations are used in solving problems related to motion, population growth, heat transfer, and chemical reactions. The applications of differential equations are used in various fields, such as physics, chemistry, engineering, and biology, in modeling and analyzing phenomena such as fluid dynamics, electrical circuits, radioactive decay, and predator-prey interactions.