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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 trickier…linear 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 θ dt 2 + g L sin θ = . Linear or nonlinear? But sin θ ≈ θ when θ is small. (Why?) So the equation becomes d 2 θ dt 2 + g L θ = , which is linear. θ L Math 2860: Summer 2010 Page 6 of 32 What is a solution? From the text: Any function φ , 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. Example: dy dt = 3 x − y dx dt = 2 x + 3 y Cannot solve one at a time because they are linked; must solve together simultaneously. ODE’s or PDEs? Math 2860: Summer 2010 Page 8 of 32 Explicit vs. Implicit Solutions Explicit Solution Has the form y = φ ( x ). Implicit Solution Has the form G ( x, y ) = 0. There must be at least one function φ that satisfies both G ( x, y ) = 0 and the differential equation on an interval I ....
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This note was uploaded on 10/29/2010 for the course DEFFERENTI 2080 taught by Professor Kidnan during the Spring '10 term at UOIT.
 Spring '10
 Kidnan

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