A differential equation: An equation involving a dependent variable and its derivatives with respect to one or more independent variables.
Ordinary differential equation: A differential equation involving derivatives with respect to a single independent variable.
Examples:
\[ \cos^2 x \frac{dy}{dx} + y = \tan x \] \[ \frac{d^3 y}{dx^3} + 6 \frac{d^2y}{dx^2} + 11 \frac{dy}{dx} + 6y = 0 \]Partial Differential Equation (PDE)
A partial differential equation is a differential equation involving derivatives with respect to two or more independent variables.
Examples:
\[ x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = 2z \] \[ \frac{\partial^3 u}{\partial x^3} + 3 \frac{\partial^3 u}{\partial x^2 \partial y} + 3 \frac{\partial^3 u}{\partial x \partial y^2} + \frac{\partial^3 u}{\partial z^3} = x^2 + y^2 + z^2 \]- The order of a differential equation is the order of the highest derivative occurring in it.
- The degreeof a differential equation is the degree (power) of the highest-order derivative after the derivatives have been made free from radicals and fractions.
- A solution of a differential equation is a relation between the dependent and independent variables (excluding derivatives), which satisfies the differential equation.
- The General (Complete) Solution of a differential equation is a solution in which the number of arbitrary constants is equal to the order of the differential equation.
- A particular solution of a differential equation is obtained from the general solution by giving particular values to the arbitrary constants
E.g., \( y = A \cos x + B \sin x \) is the general solution of \( \frac{d^2 y}{dx^2} + y = 0 \).
E.g., \( y = 2 \cos x - 3 \sin x \) is a particular solution of \( \frac{d^2 y}{dx^2} + y = 0 \).
Differential Equations of First Order and First Degree
An equation of the form \( \frac{dy}{dx} = f(x, y) \) or \( M(x, y) dx + N(x, y) dy = 0 \) is said to be a differential equation of the first order and first degree.
In this chapter, we will discuss certain types of first-order and first-degree differential equations for which solutions can be readily obtained by the following standard methods: