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Unformatted text preview: Math 2860: Summer 2010 Page 1 of 32 What is a Differential Equation??? An equation that involves the derivatives of an unknown function of one or more variables. Math 2860: Summer 2010 Page 2 of 32 Types of Differential Equations Ordinary Differential Equation (ODE) The unknown function is a function of one variable. Partial Differential Equation (PDE) The unknown function is a function of more than one variable. Math 2860: Summer 2010 Page 3 of 32 Order of a Differential Equation The order of the highest derivative involved. y + (5 y ) 7 + y = e x y (4) + x 4 y = 2 u xx + tu = 3 u tx Normal Form (ODE) Put the highest derivative on one side of the equation, and everything else on the right. (Makes it easy to see the order.) Math 2860: Summer 2010 Page 4 of 32 Linear vs. Nonlinear Differential Equations Linear The unknown function and all of its derivatives only appear with power 1 or 0, and the coefficients are all in x . Nonlinear Not linear y 3 y cos x = x 2 y 3 y 2 cos x = x 2 y + 2( y ) 2 = 1 x 3 y + x 2 y = x 4 So, a general linear ODE of order n has the form: a n ( x ) y ( n ) + a n 1 ( x ) y ( n 1) + ... + a 1 ( x ) y + a ( x ) y = g ( x ) Math 2860: Summer 2010 Page 5 of 32 Which is trickierlinear or nonlinear? Generally, nonlinear is trickier to solve (find the unknown function). Linearization is a common technique. Example: The motion of a pendulum is governed by d 2 dt 2 + g L sin = . Linear or nonlinear? But sin when is small. (Why?) So the equation becomes d 2 dt 2 + g L = , which is linear. L Math 2860: Summer 2010 Page 6 of 32 What is a solution? From the text: Any function , defined in an interval I and possessing at least n derivatives that are continuous on I , which when substituted into an n thorder ODE reduces the equation to an identity, is said to be a solution of the equation on the interval. In other words: Sub it in and LHS = RHS Example: For what value(s) of r is y = e rt a solution of 2 y y 3 y = 0? Math 2860: Summer 2010 Page 7 of 32 Systems of Differential Equations If a differential equation has more than one unknown function, then more than one equation is necessary to obtain a solution....
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 Spring '10
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