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Student_Notes_Part1 - 1 MATH 2120 MATHEMATICAL METHODS FOR...

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1 MATH 2120 MATHEMATICAL METHODS FOR DIFFERENTIAL EQUATIONS LECTURE NOTES 2 Introduction and Review Ordinary Differential Equations (ODEs) are equations that contain derivatives of an unknown function with respect to an independent variable. These often describe some physical situation and their solution gives some insight into the problem. The principle task in differential equations and their applications is to find all solutions of given equations and investigate their properties. Notation: Ordinary Derivatives ) x ( y dx y d y D ) x ( y ) x ( y dx y d y D ) x ( y ) x ( y dx dy Dy ) n ( n n n ) 2 ( 2 2 2 ) 1 (
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3 Definition: Order An ODE is the order of the highest derivative that appears in the equation. A general n th -order ODE has the form F(x, y, y , y , …, y (n) ) = 0 Examples g dt y d 2 2 2 nd order p dt dp 1 st order 4 Definition: Differential Operator If we rearrange the ODE so that all the terms involving the unknown function y(x) are on the left and all the terms not involving y(x) are on the right then L {y} = f(x) where L is the differential operator. Definition: Linear ODE The differential operator L is a linear operator if L {c 1 y 1 (x) + c 2 y 2 (x)} = L {c 1 y 1 (x)} + L {c 2 y 2 (x)} for all functions y 1 (x) and y 2 (x) and all constants c 1 and c 2 . In general a linear ODE can be written in the form x f x y x a x y x a x y x a 0 1 n 1 n n n .
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5 Definition: Homogeneous Linear ODE A linear ODE of the form L {y(x)} = 0 Definition: Nonhomogeneous Linear ODE A linear ODE of the form L {y(x)} = f(x) where f(x) 0 x . f(x) is called a forcing term. Definition: Nonlinear ODE An ODE of the form L {y(x)} = f(x) where L is not a linear operator. Examples 0 x sin dx dy x dx y d y 2 2 2 nd order nonlinear ODE. 0 x sin dx dy x dx y d 2 2 2 nd order linear nonhomogeneous ODE. 6 Initial Value and Boundary Value Problems In most applications you need a particular solution rather that a general one. When a function is integrated, an arbitrary integration constant is introduced. Thus, when you integrate a first order ODE, the general solution will contain one unknown arbitrary constant , which could be specified using a single condition to obtain the particular solution. For second order ODES we have two arbitrary constants , and so we require two conditions to specify the unknowns. Often these conditions come in the form of initial values, where both the value of the unknown function and its derivative are given for a given value of the variable, e.g. 1 0 0 0 K ) x ( y , K ) x ( y where x = x 0 is a given point and K 0 and K 1 are given numbers. These are called initial conditions . These, together with the ODE form an initial value problem .
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7 Some applications lead to conditions of the type y(P 1 ) = k 1 , y( P 2 ) = k 2 . These are known as boundary conditions , since they refer to the endpoints P 1 and P 2 (or boundary points P 1 and P 2 ) of an interval I on which the ODE is considered.
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