Differential Equations
Differential Equations:

Definition:
 Differential equations involve functions and their derivatives. They describe how a function's rate of change relates to its current state.
 An equation containing derivatives of one or more dependent variables with respect to one or more independent variables is a differential equation.

Types of Differential Equations:
a. Ordinary Differential Equations (ODEs):
 Involves one independent variable and derivatives of a function with respect to that variable.
 Example: $\frac{dy}{dx}=f(x,y)$.
b. Partial Differential Equations (PDEs):
 Involves multiple independent variables and derivatives of a function with respect to those variables.
 Example: $\frac{{\mathrm{\partial}}^{2}u}{\mathrm{\partial}{x}^{2}}+\frac{{\mathrm{\partial}}^{2}u}{\mathrm{\partial}{y}^{2}}=f(x,y,u,\frac{\mathrm{\partial}u}{\mathrm{\partial}x},\frac{\mathrm{\partial}u}{\mathrm{\partial}y})$.

Order of Differential Equations:
 The highest derivative present in the equation determines the order of the differential equation.
 Example: $\frac{{d}^{2}y}{d{x}^{2}}+3\frac{dy}{dx}2y=0$ is a secondorder differential equation.

Solutions of Differential Equations:

General Solution: Contains arbitrary constants that satisfy the differential equation but don't consider initial conditions.

Particular Solution: Obtained by applying initial conditions or constraints to the general solution, giving specific values to constants.


Methods to Solve Differential Equations:
a. Separation of Variables: Applies to certain ODEs, separating variables on either side of the equation before integrating.
b. Exact Differential Equations: Checks for the existence of an integrating factor to make the equation exact.
c. Solving Linear Differential Equations: Uses integrating factors, methods for firstorder linear equations, and other techniques for higherorder linear equations.

Applications:

Physics: Describes physical phenomena like motion, heat transfer, quantum mechanics, and more.

Engineering: Used in various engineering disciplines to model systems, circuits, vibrations, etc.

Biology, Economics, and Chemistry: Applied to model population growth, economic behavior, chemical reactions, and more.


Importance:

Fundamental in scientific and engineering fields, providing tools to model and predict behaviors of various systems.

Essential in understanding dynamic systems, providing insight into how quantities change over time or space.
