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# MATH-M343/S343 Section 3.1 Notes MATH-S343

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This 3 page Class Notes was uploaded by Kathryn Brinser on Tuesday October 18, 2016. The Class Notes belongs to MATH-S343 at Indiana University taught by Michael Jolly in Fall 2016. Since its upload, it has received 3 views. For similar materials see Honors Differential Equations in Mathematics at Indiana University.

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Date Created: 10/18/16

S343 Section 3.1 Notes- Homogeneous Second Order Equations with Constant Coefficients 9-27-16 ???? ???? ???????? Second order ODE- takes form ????????2 = ????(????,????, ????????) with independent variable ???? (or ????) and dependent variable ???? o ODE linear if ????(????,????, ????????) = ???? ???? − ???? ???? ( )???????? − ???? ???? ???? (if ???? is linear in ???? and) ???????? ???????? ???????? o ????,????,???? are specified functions of ???? but do not depend on ???? ′′ ′ Second order linear equation- takes form ???? + ???? ???? ???? + ???? ???? ???? = ???? ???? ; rewritten form of ???? ???? ???????? ????????2 + ???? ???? ) ????????+ ???? ???? ???? = ???? ???? ( ) o Given ???? ???? ???? + ???? ???? ???? + ???? ???? ???? = ???? ???? , if ???? ???? ≠ 0, divide all terms by it to get correct form o Homogeneous if ???? ???? = 0, nonhomogeneous otherwise Homogeneous with constant coefficients- ???????? + ???????? + ???????? = 0 ???????? o To solve, look for solution of form ???? = ???? (convenient form because derivatives repeat themselves with only a factor of ???? as a difference, so they can be factored out) ′ ???????? ???? = ???????? ???? = ???? ????????( ????????) = ???? ???? ???????? ???????? + ???????? + ???????? = ???????? ???? 2 ???????? + ???????????? ???????? + ???????? ????????= 0 ???????? 2 ???? (???????? + ???????? + ???? = 0 ???? ????????≠ 0 for all ????, so ???????? + ???????? + ???? = 0; known as characteristic equation o Find correct value(s) of ???? by finding roots of characteristic equation o 3 possibilities for roots1???? 2???? : Real and distinct ???? 1 ???? 2) Real and repeated ???? =1???? 2) ( ) Complex ???? = 1 + ????????,???? = ???? 2 ???????? o Solutions to equation: ???? 1 ???? ????1????, 2 = ???? ????2???? o For any ???? ,???? , general solution that also works is ???? = ???? ???? + ???? ???? = ???? ???? ????1???? + ???? ???? 2 ????(linear 1 2 1 1 2 2 1 2 combination of two known solutions), assuming ???? ,????1re2l and distinct ???? = ???? ???? ???? ????1???? + ???? ???? ???? ????2???? ′′ 1 12 1 ???? 2 2 2 ????2???? ???? = ???? ???? 1 1 + ????2????2???? Substitute these into ???????? + ???????? + ???????? = 0: ′′ ′ 2 ????1???? 2 2 ???? ????1???? 2 ???? ????1???? ????2???? ???????? + ???????? + ???????? = ???? ???? ???? ????( 1 1 + ???? 2 2 ) + ???? ???? 1 1 + ????2 2???? ) + ???? ???? 1 + ???? 2 ) = ???????? 1 ????1 1 ????+ ???????? 2 2 2 ????+ ???????? 1 1 ????1???? + ????????1???? ????1????+ ???????? 2 ????2???? = ???? ???? 1 (???????? + ???????? + ???? + ???? ???? ????2????(???????? + ???????? + ???? ) 1 ???? ???? 1 ???? ???? 2 2 2 ′′ ′ = ???? 1 1 (0 + ???? ????2 2 (0) ????1,????2known to satisfy ???????? + ???????? + ???????? = 0 = 0 ′′ ′ ∴ ???? = ???? 1 1 + ???? ???? ????2 2 a solution of ???????? + ???????? + ???????? = 0 Ex. ???? − 5???? + 6???? = 0 2 o Characteristic: ???? − 5???? + 6 = 0 (???? − 3 ???? − 2 = 0 ???? = 2,3 2???? 3???? ∴ ???? 1 ???? ,???? = ????2 are both solutions o General solution: ???? = ???? ???? 2????+ ???? ???? 3???? ′ 1 2 o Let ???? 0 = 4, ???? 0 = −1 be initial values (must be at same ???? value): ???? 0 = 4 = ???? ???? + ???? ???? 0 1 2 ????1+ ???? =24 ???? ???? =) ???? (???? ???? 2???? + ???? ???? 3????) ???????? 1 2 = ???? 1???? 2????) + ????23???? 3????) 2???? 3???? ′ = 2???? 1 + 3????2????0 0 ???? 0 = −1 = 2???? ???? + 1???? ???? 2 2???? + 3???? = −1 1 2 Solve system of equations: ????1+ ????2= 4 → ???? =14 − ???? 2 2 4 − ???? )+ 3???? = −1 2 2 8 − 2????2+ 3???? 2 −1 ????2+ 8 = −1 ???? = −1 − 8 = −9 2 ????1= 4 − −9 = 13 ∴ particular solution to IVP is ???? = 1− 9???? 3???? As ???? → ∞, negative coefficient in front of “faster” exponential dominates, ???? → −∞ Ex. Solve the initial value problem ???? + 4???? + ???? = 0, ???? 0 = 0, ???? 0 = −1. o ???? + 4???? + 1 = 0 ???? = −4±√16−4 1 1( = −4± √2 = −4±2 √ = −2 ± 3√ real and distinct 2 2 2 General solution: ???? =1???? ????2+ √ ) + ????2????(−2−√3)???? 0 0 o ???? 0 = 0 = ???? ????1+ ???? ???? 2 ????1+ ???? 2 0 ′ (2+ √ ) (2− √ ) o ???? ???? = −2(+ 3 ???? √ ) 1 + (2 − 3 √ ) 2 o ???? 0 = −1 = −2 ( 3 ???? ????√+ )21− 3 ????(???? √ ) 2 0 (−2 + 3√????)+1−2 ( 3 ???? =√−) 2 o Solve system: ????1+ ???? 2 0 → ???? =1−???? 2 (−2 + 3√−)( 2)+ −( − 3 √ =)−2 (2 − √ ????)+2−2(− 3 ???? √ −) 2 2???? − 3√ − 2???? − 3???? =√−1 2 2 2 2 −2 √???? =2−1 ???? = 1 = 1 2 2√3 √12 ???? = −1 1 √12 o ???? = −1 ????(−2+√3)????+ 1 ????(−2−√3)???? √12 √12 o As ???? → ∞, ???? → 0 (both exponents negative) ′′ ′ ′ Ex. Solve the initial value problem ???? + 2???? = 0, ???? 0 = 5, ???? 0 = 6. o ???? + 2???? = 0 ???? ???? + 2 = 0 ???? = −2,0 ???? = ????1????−2???? + ????2???? = ???? 1 −2????+ ????2 o ???? 0 = 5 = ???? ???? + ???? 1 2 ????1+ ???? 2 5 o ???? ???? = −2???? ???? 1 −2???? o ???? 0 = 6 = −2???? 1 o Solve system: ????1+ ???? 2 5 → ???? =25 − ???? 1 −2???? 1 6 → ???? =1−3 ????2= 5 − −3 = 8 o ???? = −3???? −2????+ 8 o As ???? → ∞, ???? → 8 Ex. Solve the initial value problem ???? + ???? − 2???? = 0, ???? 0 = ????, ???? 0 = 2. Then find ???? so that the solution approaches 0 as ???? → ∞. 2 o ???? + ???? − 2 = 0 (???? − 1 ???? + 2 = 0 ???? = −2,1 ???? = ????1???? + ???? 2 −2???? o ???? 0 = ???? = ???? + ???? ′ ????1 2 −2???? o ???? ???? = ???? 1 − 2???? ????2 o ???? 0 = 2 = ???? −12???? 2 o Solve system: ????1+ ???? 2 ???? → ???? =1???? − ???? 2 ???? − ???? − 2???? = 2 2 2 ???? − 3???? 2 2 −3???? 2 2 − ???? ????−2 ????2= 3 ????−2 3????−????+2 2????+2 ????1= ???? − 3 = 3 = 3 2????+2 ???? ????−2 −2???? o ???? = 3 ???? + 3 ???? 2????+2 o We want ????1= 0 (second term already goes to 0 due to negative exponent), s3 = 0 → ???? = −1 makes all solutions approach 0 as ???? → ∞ o When ???? ≠ −1, solution unbounded (???? → −∞ or ???? → ∞ depending on value of ????)

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