DiffEqNotes

DiffEqNotes - Notes on Ordinary Differential and Difference...

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Notes on Ordinary Differential and Difference Equations by John J. Seater June, 2002
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Page 2 of 87 ORDINARY DIFFERENTIAL EQUATIONS DEFINITIONS Def A differential equation is an equation that involves derivatives of a dependent variable with respect to one or more independent variables. Ex. Ex. Def Any function that is free of derivatives and that satisfies identically a differential equation is a solution of the differential equation. Ex. has (an implicit function of y) as a solution. Def A differential equation that involves derivatives with respect to a single independent variable is an ordinary differential equation (ODE). One that involves derivatives with respect to more than one independent variable is a partial differential equation (PDE). Def The order of a differential equation is the order of the highest-order derivative present. Ex. is of order 2. Ex. is of order 4. Def If a differential equation can be rationalized and cleared of fractions with regard to all derivatives present, the exponent of the highest-order derivative is called the degree of the
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Page 3 of 87 equation. Note : Not every differential equation has a degree. Ex. is of degree 2. Ex. has no degree. GENERAL REMARKS ON SOLUTIONS Three important questions arise in attempting to solve a differential equation: (1) Does a solution exist? (2) If a solution exists, is it unique? (3) If solutions exist, how do we find them? Although most effort concerns (3), questions (1) and (2) are logically prior and must be answered first. Thm (Existence) A differential equation y' = f(x,y) has at least one solution passing through any given point (x 0 , y 0 ) if f is continuous. Thm (Uniqueness) A sufficient condition for the solution to the differential equation y' = f(x,y), passing through a given point (x 0 ,y 0 ) to be unique is that M f/ M y be continuous. These theorems apply only to first-order equations. However, analagous theorems apply to higher-order equations: Thm (Existence and Uniqueness) The n-th order equation has a unique solution passing through the point
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Page 4 of 87 if f, f 2 , f 3 , . .., f n+1 all are continuous. When the last theorem applies, an n-th order differential equation has a solution with n arbitrary constants, called the general solution , and this solution is unique. Ex. has the general solution where the three arbitrary constants are A, B, and C. Special cases of the general solution are called particular solutions ; these are determined by the arbitrary initial conditions (x 0 , 0 1 , . .., 0 n ). Ex. The equation has the general solution However, it also has the solution which cannot be derived from the general solution and so is not a particular solution. It is called a singular solution . To understand this last example, note that
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This note was uploaded on 10/05/2009 for the course ECG 590 taught by Professor Msmorril during the Fall '08 term at N.C. State.

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DiffEqNotes - Notes on Ordinary Differential and Difference...

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