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Unformatted text preview: 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 ( t ) y ( n ) ( t ) + a 1 ( t ) y ( n 1) ( t ) + + a n 1 ( t ) y ( t ) + a n ( t ) y ( t ) = F ( t ) (1) with given coefficient functions a ( t ), , a n ( t ) and F ( t ) is said to be linear . Otherwise, the ODE is said to be nonlinear . For example, y ( t ) 2 + y ( t ) = 0, y ( t ) y 00 ( t ) + y ( t ) = 0 and y ( t ) = e y ( t ) are all nonlinear. (e) The ODE (1) is said to have constant coefficients if the coefficients a ( 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. (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 ( t ) , a 1 ( t ) , , a n 1 ( t ) , a n ( t ) and F ( t ) are reasonably smooth, bounded functions and that a ( t ) is not zero....
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This note was uploaded on 01/27/2011 for the course MATH 267 taught by Professor Yilmaz during the Spring '10 term at The University of British Columbia.
 Spring '10
 YILMAZ
 Differential Equations, Equations, Derivative

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