DIFFERENTIAL EQUATION Lecture #1
This document provides an overview of key concepts in differential equations: - A differential equation involves independent variables, dependent variables, and their derivatives. - An ordinary differential equation contains one independent variable and one or more dependent variables. - A partial differential equation contains more than one independent variable and dependent variables and their partial derivatives. - The order of a differential equation refers to the order of the highest differential coefficient. The degree refers to the power of the highest order differential coefficient. - Differential equations can be linear or non-linear depending on whether the dependent variable and its derivatives occur only in the first degree. - Differential equations can be formed by differentiating ordinary equations and eliminating arbitrary constants.
DIFFERENTIAL EQUATION Lecture #1
- 1. DIFFERENTIAL EQUATION (MT-201) SYED AZEEM INAM DIFFERENTIAL EQUATION (MT-201) LECTURE #1 DIFFERENTIAL: Sometimes we need a quick and simple estimate of the change in f(x) that results from a given change in f(x) that results from a given change in x. Write y for f(x) and suppose that the change in the independent variable x is the increment∆𝑥, so that x changes from its original value of x to the new value x +∆𝑥. The actual change in the value of y is the increment∆𝑦, computed by subtracting the old value of y from its new value ∆𝑦 = f(x +∆𝑥) – f(x) Now we compare the actual increment ∆𝑦 with the change that would occur in y if it continued to change at the fixed rate f’(x) as x changes to x +∆𝑥. This hypothetical change in y is called the differential dy=f’(x) ∆𝑥 DIFFERENTIAL EQUATIONS: An equation which involves independent variable, dependent variables and their derivatives, is called a differential equations. ORDINARY DIFFERENTIAL EQUATION: If an equation which involves only one independent variable, one or more dependent variables and their derivatives, is called and ordinary differential equation. PARTIAL DIFFERENTIAL EQUATIONS: If an equation which involves more than one independent variable, dependent variable and its partial derivatives, is called a partial differential equation. ORDER: The order of a differential equation is the order of the highest differential coefficient which occurs in the differential equation.
- 2. DIFFERENTIAL EQUATION (MT-201) SYED AZEEM INAM DIFFERENTIAL EQUATION (MT-201) DEGREE: The degree of a differential equations is the power of the highest order differential coefficient which occurs in the differential equations. LINEAR AND NON-LINEAR DIFFERENTIAL EQUATIONS: A differential equation is said to be linear differential equation of the dependent variable and all its derivative are occurring in the first degree, otherwise the differential equation is said to be non-linear. The differential equation is said to be non-linear. FORMATION OF DIFFERENTIAL EQUATIONS: The differential equation can be obtained by differentiating an ordinary equation and eliminating the arbitrary constants among them.