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Chapter_5_lecture

# Chapter_5_lecture - Classification of Ordinary Differential...

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Classification of Ordinary Differential Equations Classified by order, linearity, homogeneity, boundary conditions, and  autonomy: 2 2 2 0 x dy y dx dy y kx dx dy y kx dx d y dy y e dx dx + = + = + = + + = First order, linear, homogeneous:   First order, linear, nonhomogeneous:   First order, nonlinear, nonhomogeneous:   Second order, linear, nonhomogeneous:  Secon ( 29 2 2 3 2 3 2 2 3 2 3 2 sin x x d y dy y y e dx dx d y d y dy a b y e dx dx dx d y d y dy a y x dx dx dx + + = + + + = + + + = d order, nonlinear, nonhomogeneous:  Third order, linear, nonhomogeneous:  Third order, nonlinear, nonhomogeneous:

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Initial and boundary conditions To have a unique solution, an n th -order ODE requires n values of the  dependent variable (or its derivatives) to be specified at known values of  the independent variable: To have a unique solution, a set of  n first-order ODEs requires n values  of the dependent variables (or their derivatives) to be specified at known  values of the independent variable: Both of the above examples are  initial-value  problems. 2 2 0 0 0, 0, 1.5 at   x d y dy dy y y e x y dx dx dx + + = = = = 0 0 0 A 1 A 2 B B 1 A 2 B 3 B 4 C C 3 B 4 C (0) (0) (0) A A B B C C dC k C k C C C dt dC k C k C k C k C C C dt dC k C k C C C dt = - + = = - - + = = - =
Initial and boundary conditions (cont.) Boundary-value  problem: The values of the dependent variables are  fixed at more than one position of the independent variable:  0 0 A 1 A 2 B B 1 A 2 B 3 B 4 C C 3 B 4 C (0) ( ) (0) f A A B f B C C dC k C k C C C dt dC k C k C k C k C C t C dt dC k C k C C C dt = - + = = - - + = = - =

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Examples of dynamic systems 0 0 0 A 1 A 2 B B 1 A 2 B 3 B 4 C C 3 B 4 C (0) (0) (0) A A B B C C dC k C k C C C dt dC k C k C k C k C C C dt dC k C k C C C dt = - + = = - - + = = - = 3 1 2 4 A B C k k k k ÷♠ Chemical reactions in an unsteady state reactor: The component balances result in a model that is a set of simultaneous differential equations:
Autonomous & non-autonomous: A differential equation is autonomous if the independent  variable does not appear explicitly in the equation: 2 2 2 2 0 0 0 Autonomous:   Non-autonomous:   x dy y dx d y dy y dx dx dy y kx dx dy xy dx d y dy y e dx dx + = + + = + = + = + + =

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Canonical (standard) form of ODEs Consider the set of n first-order ODEs: This is an initial-value problem in its canonical form. Most integration routines require the equations to be arranged in  this form.
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Chapter_5_lecture - Classification of Ordinary Differential...

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