hw1 - Math 205, Spring 2011 B. Dodson Text - Goode-Annin,...

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Unformatted text preview: Math 205, Spring 2011 B. Dodson Text - Goode-Annin, 3rd edt. Office hours: M,F: 2:20-3:30, Tue: 11:30-12:30 and by appt. email: bad0@lehigh.edu 2 1. Course Info 2. Homework 1: Chapter 1, Sections 1, 2, 3, 4, 5, 6 Due Monday, Jan 24th Chapter 1: Intro, Slope Fields, verify solution (Monday, 1/17) 1.4, 1.5: Separable DE, Population/logistic growth (start 1.6; Wednesday, 1/19) 1.6 Linear Equations (Friday, 1/21 4 Problem 1: √ Verify that the function y = c1 x is a y solution of y ′ = 2x Solution: Compute y ′ and check. y′ = 1 1 −2 c1 ( 2 )x . √ c1 x y = 2x 2x √ x 1 = c1 ( 2 ) √ 2 ( x) 1 1 = c1 ( 2 ) √ x = y ′. Problem 2: Determine all values r so y = erx is a solution to y ′′ − 4y ′ + 3y = 0. 6 Linear DE, Main Step: to solve linear DE dy + p(x)y = q (x) dx multiply by the integral factor f = e p dx and use d (f y ) = f (y ′ + py ) dx d on the left to get (f y ) = f q. dx Problem: Solve dy 2x + y = 4x, −1 ≤ x ≤ 1. 2) dx (1 − x Solution: Find integral factor, inside integral first: 2x dx 2) (1 − x = − ln(1 − x2 ) = ln (1 − x2 )−1 (simplify!). So e 2x 1−x2 ) ( dx =e 2 −1 = 1−x Multiply by f = 1 1 − x2 ln ((1−x2 )−1 ) 1 = = f. 1 − x2 1 and use Main Property: 2 1−x 2x dy + y 2) dx (1 − x = 4x , 2 1−x 4x . 2 1−x y = (−2 ln (1 − x2 )) + c, Now integration gives 1 − x2 d dx y 1 − x2 = so y = (1 − x2 ) − ln ((1 − x2 )2 ) + c . Notice that we can check this instance of the Main Property directly (using the product rule). ...
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This note was uploaded on 11/24/2011 for the course MATH 205 taught by Professor Zhang during the Spring '08 term at Lehigh University .

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hw1 - Math 205, Spring 2011 B. Dodson Text - Goode-Annin,...

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