KNF1023 Engineering Mathematics II Introduction to ODEs Prepared By Annie ak JosephPrepared ByAnnie ak Joseph Session 2008/2009
Learning Objectives Describe the concept of ODEs Solve the problems of ODEs Apply an ODEs in real life application
Introduction to ODEs Introduction to ODEs Order of Solving an ODE ODE – general, particular, exact solutions
Basic ConceptAn ordinary differential equation is an equation with relationship between dependent variable (“y”), independent variable (“x”) and one or more derivative of y with respect to x.Example:1. y 5 x 4 ,2. y ,, xy 83. 2 x 2 y 10 y , , , , 3 xy ,, xy
Basic ConceptOrdinary Differential equations different from partial differential equationsPartial Differential equations-> unknown function depends on two or more variables, so that they are more complicated d 2V d 2V 2 2 0 dx dy
Order of ODEs:The order of a differential equation is the order of the highest derivative involved in the equation. Example: 1. y cos x , 2. y ,, 4 y 0 3. 2 x 2 y10 y ,,,, 3 xy ,, xy 4. x y y 2e y (x 2) y 2 ,,, , x ,, 2 2
Arbitrary ConstantsAn arbitrary constant, often denoted by a letter at the beginning of the alphabet such as A, B,C, c 1 , c 2 , etc. may assume values independently of the variables involved. For example in y x2 c1 x c2 , c 1 and c2 are arbitrary constants.
Solving of an Ordinary DifferentialEquationsA solution of a differential equation is a relation between the variables which is free of derivatives and which satisfies the differential equation identically.
Solving of an Ordinary DifferentialEquationsExample 1: y 6x 0 dy y , 6 xdx 3 x C 2 dx y (3 x C ) dx x Cx D 2 3
Concept of General SolutionA solution containing a number of independent arbitrary constants equal to the order of the differential equation is called the general solution of the equation.We regard any function y(x) with N arbitrary constants in it to be a general solution of N th order ODE in y=y(x) if the function satisfies the ODE.
Concept of General Solution Example 2 : y ( x) 8 x 3 Cx D is a solution for ODE y 48 x 2 d y y 2 48 x ,, dx y 48 xdx 24 x 2 C y ( 24 x 2 C ) dx 8 x 3 Cx D
Particular Solution When specific values are given to at least one of these arbitrary constants, the solution is called a particular solution. Example 3: y ( x) 8 x 3 2 x D y ( x) 8 x Cx 5 3 y ( x) 8 x 3 5 x 1
Exact SolutionA solution of an ODE is exact if the solution can be expressed in terms of elementary functions.We regards a function as elementary if its value can be calculated using an ordinary scientific hand calculator.
Exact SolutionThus the general solution y ( x) 8 x 3 Cx D of the ODE y 48 x is exact.We may not able to find exact solution for some ODEs. As example, consider the ODE dy sin( x) dx x sin( x) y dx x