lecturenotes - D I F F E R E N T I A L E Q U A T I O N S E...

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Unformatted text preview: D I F F E R E N T I A L E Q U A T I O N S E r i n P . J . P e a r s e These notes follow Differential Equations and Boundary Value Problems (4ed) by C. H. Edwards and D. E. Penney, but also include material borrowed freely from other sources. This document is not to be used for any commercial purpose. Version of September 20, 2011 Disclaimer: These notes were typed hurriedly in preparation for lecture. They are FULL of errors and are not intended for distribution to the students. Typically, the errors get corrected while proceeding through lecture on-the-fly. Caveat, caveat, caveat, etc. Contents 1 First-Order Differential Equations 1 1.1 Differential equations and mathematical models . . . . . . . . . . . . . . . . 1 1.2 Integrals as general and particular solutions . . . . . . . . . . . . . . . . . . 8 1.3 Slope fields and solution curves . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Separable equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.6 Substitution methods and exact equations . . . . . . . . . . . . . . . . . . . 26 3 Linear Equations of higher order 35 3.1 n th-Order Linear ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Homogeneous equations with constant coefficients . . . . . . . . . . . . . . . 42 ii CONTENTS Chapter 1 First-Order Differential Equations 1.1 Differential equations and mathematical models Algebra: given x 2- 3 x + 1 = 0, find the number x that solves it. DEs: given df dt = t 2 + 1, find the function f ( t ) that solves it. Definition 1.1.1. An ordinary differential equation (ODE) is an eqn involving an unknown function of one variable, and some of its derivatives. (PDEs study multivariable functions) DEs describe how deterministic systems change ; all physical laws derive from DEs. Goals of this course: 1. Recog the DE describing a given system/situation. 2. Use key characteristics/features of this DE to find an exact or approximate solution. 3. Interpret the solution: predictions, properties, etc. Differential equations are typically solved via integration , as these operations are inverses of each other. Although it may be difficult to integrate an expression, it is always easy to check your result when you are finished and the same is true for differential equations. Example 1.1.2. Check that y ( x ) = Ce x 2 solves the ODE dy dx = 2 xy . Substituting the given solution into the left side gives dy dx = d dx ( Ce x 2 ) = Ce x 2 ( d dx x 2 ) = Ce x 2 2 x = 2 x y. 2 First-Order Differential Equations To find the solution in the first place, we can use separation of variables (see 1.4): dy dx = 2 xy dy y = 2 xdx Z dy y = Z 2 xdx ln | y | + C 1 = x 2 + C 2 ln | y | = x 2 + C 3 C 3 := C 2- C 1 | y | = e x 2 + C 3 y = Ce x 2 C = e C 3 ....
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This note was uploaded on 09/26/2011 for the course MATH 3113 taught by Professor Dickey during the Fall '08 term at The University of Oklahoma.

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lecturenotes - D I F F E R E N T I A L E Q U A T I O N S E...

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