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26-Intial Value Problem

26-Intial Value Problem - ODEs IVPs ODEs IVPs Initial-Value...

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ODEs IVPs Initial-Value Problems Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Initial-Value Problems MATH 2070U 1 / 10 ODEs IVPs Initial-Value Problems 1 Ordinary differential equations (ODEs) 2 Initial-value problems (IVPs) for ODEs c D. Aruliah (UOIT) Initial-Value Problems MATH 2070U 2 / 10 ODEs IVPs Differential equations Differential equation (DE) : equality relating values of a function to values of its derivatives (rates of change) e.g., d 2 y dt 2 + ω 2 y = sin 5 t where y = y ( t ) e.g., u t = κ 2 u x 2 where u = u ( x , t ) e.g., v ( 4 ) + v = 0 where v = v ( x ) Ordinary differential equation (ODE) : only 1 independent variable Partial differential equation (PDE) : > 1 independent variable Order of differential equation : largest derivative occurring c D. Aruliah (UOIT) Initial-Value Problems MATH 2070U 4 / 10 ODEs IVPs Initial-value problems Initial-value problem Given a function f , initial data y 0 , and an interval I R , determine a function y : I R such that y = f ( t , y ( t )) , t I , y ( t 0 ) = y 0 .
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