Chapter 4

प्रस्तावना

In this section, the textbook introduces the importance of differential equations in various fields such as Physics, Engineering, and other applied sciences. Many real-world problems involve situations where independent variables, dependent variables, and their derivatives are present together. This unit will help you understand such differential equations and their relevance. The focus is on learning the basic concepts and methods to solve differential equations, especially those of the first order and first degree.

• Differential equations are widely used in Physics, Engineering, and Applied Sciences.
• Problems often involve relationships between variables and their derivatives.
• This section sets the stage for studying differential equations and their solutions.
• 📌 Differential Equation: An equation involving independent variable(s), dependent variable(s), and their derivatives.

परिभाषा

A differential equation is defined as an equation that relates the independent variable(s), the dependent variable(s), and their derivatives. In other words, it is an equation involving one or more derivatives of a function. For example: (i) \( \frac{dy}{dx} + xy = \cos x \) (ii) \( x^2 \frac{d^2y}{dx^2} + y^2 = 0 \) (iii) \( \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} = 0 \) These examples illustrate equations involving derivatives of various orders and with respect to different variables.

• A differential equation relates variables and their derivatives.
• Can involve ordinary or partial derivatives.
• Examples include both ordinary and partial differential equations.
• 📌 Independent Variable: The variable with respect to which differentiation is performed.
• 📌 Dependent Variable: The variable whose derivative is taken.
• 📌 Derivative: The rate of change of a function with respect to a variable.