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1. odeReview

# 1. odeReview - Review of Ordinary Differential Equations...

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Review of Ordinary Differential Equations Definition 1 (a) A differential equation is an equation for an unknown function that contains the derivatives of that unknown function. For example y 00 ( t ) + y ( t ) = 0 is a differential equation for the unknown function y ( t ). (b) A differential equation is called an ordinary differential equation (often shortened to “ODE”) if only ordinary derivatives appear. That is, if the unknown function has only a single independent variable. A differential equation is called a partial differential equation (often shortened to “PDE”) if partial derivatives appear. That is, if the unknown function has more than one independent variable. For example y 00 ( t ) + y ( t ) = 0 is an ODE while 2 u ∂t 2 ( x, t ) = c 2 2 u ∂x 2 ( x, t ) is a PDE. (c) The order of a differential equation is the order of the highest derivative that appears. For example y 00 ( t ) + y ( t ) = 0 is a second order ODE. (d) An ordinary differential equation that is of the form a 0 ( t ) y ( n ) ( t ) + a 1 ( t ) y ( n - 1) ( t ) + · · · + a n - 1 ( t ) y 0 ( t ) + a n ( t ) y ( t ) = F ( t ) (1) with given coefficient functions a 0 ( t ), · · · , a n ( t ) and F ( t ) is said to be linear . Otherwise, the ODE is said to be nonlinear . For example, y 0 ( t ) 2 + y ( t ) = 0, y 0 ( t ) y 00 ( t ) + y ( t ) = 0 and y 0 ( t ) = e y ( t ) are all nonlinear. (e) The ODE (1) is said to have constant coefficients if the coefficients a 0 ( t ), a 1 ( t ), · · · , a n ( t ) are all contants. Otherwise, it is said to have variable coefficients . For example, the ODE y 00 ( t ) + 7 y ( t ) = sin t is constant coefficient, while y 00 ( t ) + ty ( t ) = sin t is variable coefficient. (f) The ODE (1) is said to be homogeneous if F ( t ) is identically zero. Otherwise, it is said to be in- homogeneous or nonhomogeneous . For example, the ODE y 00 ( t ) + 7 y ( t ) = 0 is homogeneous, while y 00 ( t ) + 7 y ( t ) = sin t is inhomogeneous. A homogeneous ODE always has the trivial solution y ( t ) = 0. (g) An initial value problem is a problem in which one is to find an unknown function y ( t ) that satisfies both a given ODE and given initial conditions, like y (0) = 1, y 0 (0) = 0. (h) A boundary value problem is a problem in which one is to find an unknown function y ( t ) that satisfies both a given ODE and given boundary conditions, like y (0) = 0, y (1) = 0. Theorem 2 Assume that the coefficients a 0 ( t ) , a 1 ( t ) , · · · , a n - 1 ( t ) , a n ( t ) and F ( t ) are reasonably smooth, bounded functions and that a 0 ( t ) is not zero. (a) The general solution to the ODE (1) is of the form y ( t ) = y p ( t ) + C 1 y 1 ( t ) + C 2 y 2 ( t ) + · · · + C n y n ( t ) where n is the order of the ODE (1) the particular solution, y p ( t ) , is any solution to (1) C 1 , C 2 , · · · , C n are arbitrary constants y 1 , y 2 , · · · , y n are n

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1. odeReview - Review of Ordinary Differential Equations...

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