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Week 1 [compatibility mode]
Week 1 [compatibility mode]
Week 1 [compatibility mode]
Week 1 [compatibility mode]
Week 1 [compatibility mode]
Week 1 [compatibility mode]
Week 1 [compatibility mode]
Week 1 [compatibility mode]
Week 1 [compatibility mode]
Week 1 [compatibility mode]
Week 1 [compatibility mode]
Week 1 [compatibility mode]
Week 1 [compatibility mode]
Week 1 [compatibility mode]
Week 1 [compatibility mode]
Week 1 [compatibility mode]
Week 1 [compatibility mode]
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  • 1. KNF1023 Engineering Mathematics II Introduction to ODEs Prepared By Annie ak JosephPrepared ByAnnie 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  83. 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 DifferentialEquations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 DifferentialEquations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
  • 15. Applications of ODEs
  • 16. Summary Order of ODE Solving an ODE ODEs general, particular, exact solutions
  • 17. Prepared By Annie ak JosephPrepared ByAnnie ak Joseph Session 2008/2009

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