Differential equations

Introduction to Differential Equations

• A differential equation is a mathematical equation that involves variables and their derivatives.
• It is an equation that contains derivative of one or many unknown functions.

Classification of Differential Equations

• Ordinary differential equations: Equations that contain only one variable and its derivatives.
• Partial differential equations: Equations that contain partial derivatives, which means the functions are dependent on more than one variable.
• Linear differential equations: A type of ordinary differential equation that is represented as a linear polynomial.
• Non-linear differential equations: Any differential equation that is not a linear polynomial.

Solution of Differential Equations

• The solution to a differential equation involves finding a function or a set of functions that satisfied the given equation. This is often known as the general solution.
• A particular solution refers to a solution achieved from the general solution by applying the initial or boundary conditions.
• The process of finding solutions is known as solving or integrating the differential equation.

First Order Differential Equations

• First order differential equations contain only the first derivative of the function, but no higher order derivatives.
• Separable equations is a sub category in first order differential equations where the equation can be written as a product of two functions, one only in terms of the dependent variable and the other only in terms of the independent variable.
• The method of exact differential equations can be used to solve certain types of first order differential equations.

Second Order Differential Equations

• Second order differential equations contain a second order derivative, but no higher order derivatives.
• Homogeneous second order differential equations are those whose non-derivative part equals zero.
• The solutions to second order differential equations could be real and distinct, real and repeated, or complex conjugates depending on the nature of roots of the equation’s characteristic equation.

Applications of Differential Equations

• Differential equations are used in many areas of science and engineering to model phenomena or systems in which change occurs.
• For example, in physics, differential equations are used to model the motion of oscillating systems (like springs), electronic circuits, and planetary motion, among others.