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Chapter1Section1

# Chapter1Section1 - 1.1 Differential Equations and...

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1.1: Differential Equations and Mathematical Models Basic Definitions An ordinary differential equation is an equation that relates an unknown function, one independent variable, and one or more of the functions derivatives. A partial differential equation is a differential euqation which has two or more independent variables. Notation : F H x , y H x L , y ' H x L , y '' H x L , ..., y ^ H n LL = 0 This represents an n th order differential equation with independent variable x . y H n L H x L = G I x , y H x L , y ' H x L , ..., y I n - 1 M H x LM This is called the normal form for a n th order differential equation with independent variable x . Initial Value Problems are differential equations with extra information or initial conditions: y H a L = b 0 , y ' H a L = b 1 , y '' H a L = b 2 , ..., y I n - 1 M H a L = b n - 1 . A s o l u t i o n to F H x , y H x L , y ' H x L , y '' H x L , ..., y ^ H n LL = 0 is a function u H x L that is continuous on an interval I such that u ', u '', ..., u H n L exist on I and F H x , u H x L , u ' H x L , u '' H x L , ..., u ^ H n LL = 0 is true for all x ˛ I .

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Chapter1Section1 - 1.1 Differential Equations and...

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