Introduction to Differential Equations at Ishik University

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. 2 Faculty of Engineering Differential Equations Lecture 1 Introduction

Ishik University Introduction to Differential Equations The derivative of a function ?(?) with respect to x is; ?(?) ?? ?? , ? ? , The derivative is also denote by ? 3 Faculty of Engineering Differential Equations Lecture 1 Introduction

Ishik University Introduction to Differential Equations Example 2 x = = ( ) y f x e 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. 2 x ( ) 2 ( ) dy d e d x 2 2 x x = = = = 2 2 e e y dx dx dx 4 Faculty of Engineering Differential Equations Lecture 1 Introduction

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

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) 6 Faculty of Engineering Differential Equations Lecture 1 Introduction

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

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

Ishik University Classification of Differential Equations Ordinary Differential Equation (ODE) This type of differential equations contains only ONE independent variable x, so the dependent variable y is completely depend on x. ?? ??= ???? ?????? ?? ??= ????? + ?? 9 Faculty of Engineering Differential Equations Lecture 1 Introduction

Ishik University Classification of Differential Equations Partial Differential Equation (PDE) This type of differential equations depends on TWO or more independent variable x, y, etc., so the dependent variable u is partially depend on x and partially depend on y, etc. ?? ??+?? ??= ? ?? ??+?? ??+ ? = ?? 10 Faculty of Engineering Differential Equations Lecture 1 Introduction

Ishik University Classification of Differential Equations Differential Equations are Classified by; 1. Type, 2. Order, The order of a differential equation is the order of the highest derivative found in the DE. ??? ???+ ??? ?? ?? = ???? + ??? 11 Faculty of Engineering Differential Equations Lecture 1 Introduction

Ishik University Classification of Differential Equations The order of a differential equation is the order of the highest derivative found in the DE. ??? ???+ ??? ?? ?? = ???? + ??? First degree Second degree This equation is a second order ordinary differential equation 12 Faculty of Engineering Differential Equations Lecture 1 Introduction

Ishik University Classification of Differential Equations The order of a differential equation is the degree of the highest derivative found in the DE. ??? ???+ ??? ?? ?? = ???? + ??? First degree Second degree This equation is a second order ordinary differential equation 13 Faculty of Engineering Differential Equations Lecture 1 Introduction

Ishik University Classification of Differential Equations 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 ' ( ) n = ( , , ,...... ) 0 F x y y y 14 Faculty of Engineering Differential Equations Lecture 1 Introduction

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 15 Faculty of Engineering Differential Equations Lecture 1 Introduction

Ishik University Classification of Differential Equations dy + y = 0 is linear dt dx + x 2= 0 is non-linear dt dy +t 2= is linear 0 dt dy + t 2= 0 y is non-linear dt 16 Faculty of Engineering Differential Equations Lecture 1 Introduction