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# MATH-M343/S343 Exam 1 STUDY GUIDE MATH-S343

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This 5 page Study Guide was uploaded by Kathryn Brinser on Wednesday September 21, 2016. The Study Guide belongs to MATH-S343 at Indiana University taught by Michael Jolly in Fall 2016. Since its upload, it has received 12 views. For similar materials see Honors Differential Equations in Mathematics at Indiana University.

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Date Created: 09/21/16

S343 Exam 1 (Sections 1.1-2.8) Study Guide 9-20-16 General Points Linear o Integrating factor Nonlinear o Separable o Exact o Bernoulli If exact is method to use, will not have to manipulate with integrating factor to make equation exact (ie. given equation will be exact) All separable equations are exact, but not all exact equations are separable o Separable- ???? ???? ???????? = ???? ???? ???????? 1. ???????? ???????? a. Solve ???????? = ????????+???? , ???? ???? = −???? explicitly to find the unique solution. Nonlinear: separable ????????(2???? + 2 = ???? ???? ???????? (2???? + 2 ???????? = ???? ???????? ∫ 2???? + 2???????? = ∫ ???? ???????? 2 ???? ???? + 2???? = ???? + ???? (−4 ) + 2 −4 − ???? = ???? 16 − 8 − 1 = ???? 7 = ???? ???? + 2???? − ???? = 7 ???? + 2???? − ???? − 7 = 0 2 ???? ???? + 2???? − ???? + 7 = 0 −2± 2 −4 1 −???? −7) ???? = 2 −2±√4+4???? +28 = 2???? = −2±√4???? +32 2 ???? = −2±2√???? +8 2 = −1 ± ????√+ 8 0 −4 = −1 ± ????√+ 8 = −1 ± 9√ = −1 − 3| | ∴ ???? ???? = −1 − ????√+ 8 ???????? −????−???????? b. Solve = ???? , ???? ???? = ???? explicitly to find the unique solution. ???????? ???????? ???? +???????? (ln ???? + 2????)???????? = −???? − 2???? ???????? ???? ????+ 2???? + ln ???? + 2???? )???????? = 0 ???? ???????? ???? ( + 2????) = 1 ???????? ???? ???? ???? (ln ???? + 2???? = 1 ???????? ???? Nonlinear: exact ???? ????????= ???? = + ???????? ???? = ????+ 2???????????? ∫ ???? 2 = ????ln ???? + ???? + ℎ(????) ???? = ???? = ln ???? + 2???? ???? ???? = ∫ ln ???? + 2???????????? = ????ln ???? + ???? + ???? ???? ( ) 2 2 ???? ????,???? = ????ln ???? + ???? + ???? = ???? 2ln 1 + 1 + 2 = ???? 0 + 1 + 4 = ???? 5 = ???? ????ln ???? + ???? + ???? = 5 2 2 ????ln ???? + ???? + ???? − 5 = 0 ???? + ????ln ???? + ???? − 5 = 0 ) −ln ???? ± ln ????|−4 1 ???? −5 ) ???? = 2 −ln ???? ± ln ???? −4???? +20 = 2 −ln 1 ± ln 1 −4 1 +20 2 = 0± 0−4+20 = √ 2 = ± √16 2 4 | = + 2 −ln ???? + ln ???? −4???? +20 ∴ ????(????) = 2 2. ???????? ???????? ????????????(???? )???????? a. Solve ???????? = ???? explicitly for ???? = ???? ???? to find its general solution. ???????? 4 2 3???? ????????= 2???? cos ????( ) + ???? ???????? 3 4 2 ???????? − ???? = 2???? cos ???? ( ) Linear −3 Let ???? ???? = ???? ∫???? ???????? = ???? −3ln???? ln????−3 = ???? = ???? −3 −3 ???????? 3 −3 4 −3 2 ???? ????????− ???? (???? )???? = 2???? ????( )cos ???? ) −3???????? −4 2 ???? ???????? − 3???? ???? = 2????cos ????( ) ???? −3 2 ???????? (???? ???? = 2????cos ???? ( ) ???? −3 2 ∫ ???????? ???? ???? = ∫ 2????cos ???? )???????? −3 ( 2 ) ???? ???? = sin ???? 3 + ???? 2 3 ???? = ????(????) = ???? sin ????( )+ ???????? ???????? ????−???????? b. Solve ???????? = ???????? explicitly for ???? = ???? ???? to find its general solution. ???????? ???? = 3− ???? ???????? ???? = ???????? −3 − ???? ???????? −3 ???????? + ???? = ???????? Nonlinear: Bernoulli ( ) Let ???? = ????1−???? = ???? 1− −3 = ???? 4 ???????? ???? ????????= ????????(????) ???? 4 = ???????? (???? ) 3???????? = 4???? ???????? = 4???? ???????? −3− ???? ) 4 = 4???? − 4???? = 4???? − 4???? ???????? + 4???? = 4???? ???????? Let ???? ???? = ???? ∫ 4????????= ???? 4???? 4???????????? 4???? 4???? ???? ???????? + 4???????? = 4???????? ???? 4???? 4???? ???????? (4???????? )= 4???????? 4???????? 4???? = ∫ 4???????? 4???????????? 4???? 4???? 4???? Integration by part∫: ???????? ???????? Let ???? = ????, ???????? = ????????, ???????? = 4???? ????????, ???? = ???? ∫ ????????4???????????? = ???????? 4????− ∫???? ????????4???? = ???????? 4???? − ????4???? + ???? 4???? 4???? 4???? 4???????? = ???????? − ???? + ???? ???????? 4???? = ???????? 4????− ???? 4???? + ???? 1 4 1 4 ???? = ???? − + ???????? −4???? 4 1 1 ???? = ???????? −4???? + ???? − 4 4 1 ( ) −4???? 1 1 4 ???? = ???? ???? = (???????? + 4 − )4 3. Given that ???????? = ???????? + ???? and ???? ???? = ????: ???????? a. Approximate ???? ????.???? and ???? ????.???? using Euler’s method and ???? = ????.????. ???? = ???? + ℎ ???? ???? ,???? ) 1 0 ( 0 0 ) = 4 + 0.1 2 2 + 4 ) = 4 + 0.1 4 + 4 ) = 4 + 0.1 8) = 4.8 ≈ ???? 2.1 ) ???? 2 ???? +1ℎ ???? ( ,????1 1)) ( ( ) ) = 4.8 + 0.1 2 2.1 + 4.8 = 4.8 + 0.1 4.2 + 4.8) = 4.8 + 0.1 9) = 5.7 ≈ ???? 2.2 ) ???????? b. Rewrite the initial value problem in an equivalent way so that it has the fo= ???? ????,???? and ???????? satisfies ???? ???? = ????. ???? ???? = ???? ???? + 2 ) ???? 0 = ???? 0 + 2 ) ( ) = ???? 2 = 4 does not satisfy ???? 0 = 0 ???? ???? = ???? ???? + 2 − 4 ???? 0 = ???? 2 − 4 = 4 − 4 = 0 satisfies ???? 0 = 0 ???????? ???????? ???????? (???? = ????????(???? + 2) = 2 ???? + 2 + ???? ???? + 2 ) ( ( ) ) = 2???? + 4 + ???? ???? + 4 ???????? = 2???? + ???? + 8 notice similarity towith shift of 8 ???????? ???????? 4. Apply Picard iterations to find the first three approximations f= ???????? + ????, ???? ???? = ????. ???? ???????? ???? ????+1(???? = ∫ ???? ????(???? ???????? ( ))???????? 0 ???? ???? = ∫???? 2???? + 0???????? 1 0 2???? = ???? 0 = ????2 ???? ????2???? = ∫0 2???? + ???? ???????? 1 ???? = ???? + ???? | 3 3 0 = ???? + ????1 3 3 ???? ???? = ∫???? 2???? + ???? + ???? ???????? 3 0 3 2 1 3 1 1 4???? = ???? + ????3+ ( )????3 4 ????3 ????4 0 = ???? + + 3 12 5. Find the largest possible interval of existence for the solutions to the following problems: ???????? ???? a. ???????? = ???????????? , ???? ???? = ???? Nonlinear: separable −2 ???? ???????? = 2???? ???????? ∫ ????−2 ???????? = ∫ 2???????????? −1 2 −???? −1 ???? + ???? ???? = −1???? +???? 1 = 2 0 +−1 1 = ???? ???? = −1 ( ) −1 ???? ???? = ???? −1 Not continuous at ???? = ±1 Maximal interval of existence is −1,1 because ????0= 0 ???? −1,1 ) b. ???????? = ???????????? , ???? ???? = −???? (compare the interval to part a) ???????? −1 = −1 0 +???? ???? = 1 −1 ???? ???? = ???? +1 Never discontinuous; maximal interval of existence is −∞,∞ c. (???????????????? )????????+ ???? ???? = ???? + ????, ???? ???? = ???????????? ???????? Linear; do not have to solve to answer ???????? ???????? ???? + 1 + ???? = ???????? cos???? cos???? ???????? ???? ???? 3???? 5???? ???? ???? = cos???? not continuous at odd multiples 2f : 2 ,± 2,± 2, etc. ???? +1 ???? ???? = cos???? same as ???? ????) −???? ???? Because ???? 0 0, ( 2 ,2) is maximal interval of existence 6. Find an integrating factor ???? for each of the following equations (do NOT solve): a. ???? ???? + ???????????????? + ???? ???????? = ????′ ???? = 2???? ????, ???? = 2???????? ???? ???? Not exact, as expected ???? −???? 2???? ????−2???????? ???? ????= ???? ???? ???? 2????????(???? −1) = 2 2???? ???? 2???? −2 = ???? 2 = 2???? − ???? function of ???? as it should be ∴ ???? = ???? ???? ) ???????? 2 ???????? = (2???? − )???? 1 2 ???????? = (2???? − )???????? ???? ???? 1 2 ∫???? ???????? = ∫2???? − ???????? 2 ln ???? = ???? − 2ln ???? + ???? ???? = ???????? −2ln ???? +???? 2 = ???? ???? −2ln ???????????? 2 = ???????? −2???????? ???? ???? b. ???? + ???? + ( + ???????? + ????)???? = ???? ???? ???? ???? 1, ???? =???????? ???? + ???? + 12 Not exact ????????−???? ???? 1−???? ????+???? +1 = ???? ???? ???? +???????? +???? 23 2 = ???? −???? ???? ???? ????+???????? +???? 3 ????(????−????2) = ???? ???? 2 ????( 3 +???? +1) = no ???? −???? 2 2 ???? ????= ???? ????+???? +1−1 ???? ???? +???? ???? ????+???? 2 = 2 ???? +???? ????(???? +????) = ???? +???? = ???? ∴ ???? = ???? ???? ) ???????? = ???????? ???????? 1 ???????????? = ???? ???????? 1 ∫ ???????? = ∫???????????? ???? ln ???? = ???? + ???? 2 ???? +???? ???? = ???? 2 1 2 = ????2???? ???? ???? 1 2 = ???????? 2????

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