Chapter 10

प्रस्तावना

In the introduction, it is explained that in physics, engineering, and other branches of applied sciences, situations often arise where problems involve independent variables, dependent variables, and their derivatives. This unit will help you study such differential equations, understand their formation, and methods of solution. The focus is on the foundational understanding of differential equations, especially those of first order and first degree, and their application in various scientific problems.

• Differential equations are crucial in modeling scientific and engineering problems.
• They involve relationships among variables and their derivatives.
• This section sets the stage for understanding and solving differential equations.
• 📌 Differential Equation: An equation involving derivatives of a function.

परिभाषा

A differential equation is defined as an equation that relates an independent variable, a dependent variable, and derivatives of the dependent variable with respect to the independent variable. For example, equations involving dy/dx, d²y/dx², etc., are differential equations. The section provides examples to illustrate this concept: (i) \( \frac{dy}{dx} + xy = \cos x \) (ii) \( x^2 \frac{d^2y}{dx^2} + y^2 = 0 \) (iii) \( \frac{d^2z}{dx^2} + \frac{d^2z}{dy^2} = 0 \) These examples show how differential equations can involve one or more derivatives and variables.

• Differential equations connect variables and their derivatives.
• They are foundational in mathematical modeling.
• Examples include both ordinary and partial derivatives.
• 📌 Independent Variable: The variable with respect to which differentiation is performed.
• 📌 Dependent Variable: The variable whose value depends on the independent variable.
• 📌 Derivative: The rate of change of the dependent variable with respect to the independent variable.