Week 1 [compatibility mode]
- 1. KNF1023 Engineering Mathematics II Introduction to ODEs Prepared By Annie ak Joseph Prepared By Annie ak Joseph Session 2008/2009
- 2. Learning Objectives Describe the concept of ODEs Solve the problems of ODEs Apply an ODEs in real life application
- 3. Introduction to ODEs Introduction to ODEs Order of Solving an ODE ODE – general, particular, exact solutions
- 4. Basic Concept An 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 8 3. 2 x 2 y 10 y , , , , 3 xy ,, xy
- 5. Basic Concept Ordinary Differential equations different from partial differential equations Partial 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
- 6. 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
- 7. Arbitrary Constants An 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.
- 8. Solving of an Ordinary Differential Equations A solution of a differential equation is a relation between the variables which is free of derivatives and which satisfies the differential equation identically.
- 9. Solving of an Ordinary Differential Equations Example 1: y 6x 0 '' dy y , 6 xdx 3 x C 2 dx y (3 x C ) dx x Cx D 2 3
- 10. Concept of General Solution A 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.
- 11. 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
- 12. 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
- 13. Exact Solution A 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.
- 14. Exact Solution Thus 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
- 16. Summary Order of ODE Solving an ODE ODEs general, particular, exact solutions
- 17. Prepared By Annie ak Joseph Prepared By Annie ak Joseph Session 2008/2009