INTRODUCTION TO DIFFERENTIAL EQUATION PRESENTED BY : ANASTACIO G. PANTALEON, JR. WIDE APPLICATIONS OF DIFFERENTIAL EQUATION Economics Mechanics Physics Engineering Chemistry Biology Crime Analytics Social Science LANGUAGE OF THE DIFFERENTIAL EQUATION DEGREE OF ODE ➢ ORDER OF ODE ➢ SOLUTIONS OF ODE ❖ GENERAL SOLUTION ❖ PARTICULAR SOLUTION ❖ TRIVIAL SOLUTION ❖ SINGULAR SOLUTION ❖ EXPLICIT AND IMPLICIT SOLUTION ➢ HOMOGENEOUS EQUATIONS ➢ NON-HOMOGENEOUS EQUTIONS ➢ INTEGRATING FACTOR ➢ DEFINITION A Differential Equation is an equation containing the derivative of one or more dependent variables with respect to one or more independent variables. For example, CLASSIFICATION Differential Equations are classified by : Type, Order, Linearity, Classifiation by Type: Ordinary Differential Equation If a Differential Equations contains only ordinary derivatives of one or more dependent variables with respect to a single independent variables, it is said to be an Ordinary Differential Equation or (ODE) for short. For Example, Partial Differential Equation If a Differential Equations contains partial derivatives of one or more dependent variables of two or more independent variables, it is said to be a Partial Differential Equation or (PDE) for short. For Example, Classifiation by Order: The order of the differential equation (either ODE or PDE) is the order of the highest derivative in the equation. For Example, Order = 3 Order = 2 Order = 1 General form of nth Order ODE is = f(x,y,y1,y2,….,y(n)) where f is a real valued continuous function. This is also referred to as Normal Form Of nth Order Derivative So, when n=1, when n=2, = f(x,y) = f(x,y,y1) and so on … CLASSIFICATIONS BY LINEARITY Linear (n) The n th Order ODE is said to be linear if F( x , y , y , y ,......, y ) = 0 is linear in y 1 , y 2 , ......., y n In other words, it has the following general form: dny an ( x) dx n + an−1( x ) now for n = 1, and for n = 2, d n−1 y dx n−1 + ...... + a 2 ( x ) d2y dx 2 + a 1 ( x ) dy + a 0 ( x ) y = g ( x ) dx a 1 ( x ) dy + a 0 ( x ) y = g ( x ) dx d2y + a 1 ( x ) dy + a 0 ( x ) y = g ( x ) a2( x) 2 dx dx Non-Linear : A nonlinear ODE is simply one that is not linear. It contains nonlinear functions of one of the dependent variable or its derivatives such as: siny ln y ey Trignometric Exponential Logarithmic Functions Functions Functions LINEAR For Example, ( y − x ) dx + 5 x dy = 0 y − x + 5 xy = 0 5 xy + y = x st which are linear 1 Order ODE Likewise, Linear 2nd Order ODE is y − 5 xy + y = 2 x Linear 3rd Order ODE is y + xy − 5 y = e 2 x Non-Linear For Example, (1 − y )y + 5 y = e y + cos y = 0 y (4 ) + y 2 = 0 x SUMMARY ON THE CLASSIFICATION OF DIFFERENTIAL EQUATION ➢Type: Ordinary Partial ➢Order : 1st, 2nd, 3rd,....,nth ➢Linearity : Linear Non-Linear METHODS AND TECHNIQUES COVERED BY OUR SUBJECT • ➢VARIABLE SEPARABLE FORM • ➢VARIABLE SEPARABLE FORM, BY SUITABLE SUBSTITUTION • ➢HOMOGENEOUS DIFFERENTIAL EQUATION • ➢HOMOGENEOUS DIFFERENTIAL EQUATION, BY SUITABLE SUBSTITUTION (I.E. NON-HOMOGENEOUS DIFFERENTIAL EQUATION) ➢EXACT DIFFERENTIAL EQUATION ➢EXACT DIFFERENTIAL EQUATION, BY USING INTEGRATING FACTOR ➢LINEAR DIFFERENTIAL EQUATION ➢LINEAR DIFFERENTIAL EQUATION, BY SUITABLE SUBSTITUTION ➢BERNOULLI’S DIFFERENTIAL EQUATION ➢METHOD OF UNDETERMINED CO-EFFICIENTS ➢METHOD OF REDUCTION OF ORDER ➢METHOD OF VARIATION OF PARAMETERS ➢SOLUTION OF NON-HOMOGENEOUS LINEAR DIFFERENTIAL EQUATION HAVING NTH ORDER • • • • • • • • • • • PROBLEM In a certain House, a police were called about 3’O Clock where a murder victim was found. Police took the temperature of body which was found to be34.5 C. After 1 hour, Police again took the temperature of the body which was found to be 33.9 C. The temperature of the room was 15 C So, what is the murder time? Sir Issac Newton “ THE RATE OF COOLING OF A BODY IS PROPORTIONAL TO THE DIFFERENCE BETWEEN ITS TEMPERATURE AND THE TEMPERATURE OF THE SURROUNDING AIR ” TEMPERATURE(ф) TIME(T) First Constant Ф = 34.5OC Second Constant Ф = 33.9OC 1. The temperature of the room 15OC 2. The normal body temperature of human being 37OC MATHEMATICALLY, EXPRESSION CAN BE WRITTEN AS – d ( − 15 .0 ) dt d dt = k ( − 15 .0 ) where ' k' is the constant d = k .dt ( − 15 .0 ) ln ( − 15 .0 ) = k.t where ' c' is the of proportion .... (Variable ality Separable +c constant of integration Form ) ln (34.5 -15.0) = k(0) + c c = ln19.5 ln (33.9 -15.0) = k(1) + c ln 18.9 = k+ ln 19.5 k = ln 18.9 - ln 19.5 = - 0.032 = -0.032t + ln 19.5 ln (Ф -15.0) Substituting, Ф = 37OC ln22 = -0.032t + ln 19.5 t = ( ln 2 2 − ln 1 9 . 5 ) − 0 .032 = − 3 . 8 6 h ou r s = − 3 h ou r s 5 1 m in u te s So, subtracting the time four our zero instant of time i.e., 3:45 a.m. – 3hours 51 minutes i.e., 11:54 p.m.