Differential Equations Lecture 1 Introduction to Differential Equations

Differential Equations Lecture 1 Introduction to Differential Equations Instructor: A. S. Brwa / MSc. In Structural Engineering College of Engineering / Ishik University

Ishik University Introduction to Differential Equations DEFINITION: Differential Equation: An equation containing the derivative of one or more dependent variables, with respect to one or more independent variables is said to be a differential equation (DE). Solving a differential equation means finding the value of the dependent variable in terms of the independent variable. Faculty of Engineering – Differential Equations – Lecture 1 – Introduction 2

Ishik University Introduction to Differential Equations � Faculty of Engineering – Differential Equations – Lecture 1 – Introduction 3

Ishik University Introduction to Differential Equations Example �The dependent variable is y �The independent variable is x This example use y as the dependent variable, so the goal in each problem is to solve for y in terms of x. Faculty of Engineering – Differential Equations – Lecture 1 – Introduction 4

Ishik University Introduction to Differential Equations was first invented by English physicist Isaac Newton and German mathematician Gottfried Leibniz Faculty of Engineering – Differential Equations – Lecture 1 – Introduction 5

Ishik University The Use of Ordinary Differential Equations �All branches of Engineering �Economics �Biology and Medicine �Chemistry, Physics etc Anytime you wish to find out how something changes with time (and sometimes space) Faculty of Engineering – Differential Equations – Lecture 1 – Introduction 6

Ishik University Classification of Differential Equations are Classified by; 1. Type, 2. Order, 3. Linearity. Faculty of Engineering – Differential Equations – Lecture 1 – Introduction 7

Ishik University Classification of Differential Equations are Classified by; 1. Type, There are two main types of differential equation: 1. Ordinary 2. Partial Faculty of Engineering – Differential Equations – Lecture 1 – Introduction 8

Ishik University Classification of Differential Equations � Faculty of Engineering – Differential Equations – Lecture 1 – Introduction 9

Ishik University Classification of Differential Equations � Faculty of Engineering – Differential Equations – Lecture 1 – Introduction 10

Ishik University Classification of Differential Equations � Faculty of Engineering – Differential Equations – Lecture 1 – Introduction 11

Ishik University Classification of Differential Equations � Second degree First degree Faculty of Engineering – Differential Equations – Lecture 1 – Introduction 12

Ishik University Classification of Differential Equations � Second degree First degree Faculty of Engineering – Differential Equations – Lecture 1 – Introduction 13

Ishik University Classification of Differential Equations are Classified by; 1. Type, 2. Order, 3. Linearity. An n-th order differential equation is said to be linear if the function Faculty of Engineering – Differential Equations – Lecture 1 – Introduction 14

Ishik University Classification of Differential Equations �The important issue is how the unknown y appears in the equation. A linear equation involves the dependent variable (y) and its derivatives by themselves. There must be no "unusual" nonlinear functions of y or its derivatives. �A linear equation must have constant coefficients, or coefficients which depend on the independent variable (t). If y or its derivatives appear in the coefficient the equation is non-linear Faculty of Engineering – Differential Equations – Lecture 1 – Introduction 15

Ishik University Classification of Differential Equations is linear is non-linear Faculty of Engineering – Differential Equations – Lecture 1 – Introduction 16