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# MATH-M343/S343 Section 2.6 Notes MATH-S343

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This 5 page Class Notes was uploaded by Kathryn Brinser on Wednesday September 21, 2016. The Class Notes belongs to MATH-S343 at Indiana University taught by Michael Jolly in Fall 2016. Since its upload, it has received 2 views. For similar materials see Honors Differential Equations in Mathematics at Indiana University.

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Date Created: 09/21/16

S343 Section 2.6 Notes- Exact Equations and Integrating Factors 9-8-16 ???? Take form (????(????,???? ???? ))= 0 ???????? o Now suppose ???? = ???? ???? ,???? = ???? ????( ) o ???? (???? ( ???? ,???? ???? ))= 0 ???????? o ???????? ????????+ ???????? ???????? = 0 ???????? ???????? ???????? ???????? o Now let ???? = ???? ???????? ???????? ???????? ???????? o ???????? ????????+ ???????? ????????= 0 ???????? ???????? ???????? o ???????? + ???????? ????????= 0 ???????? o ???? ????,???? + ???? ????,???? )???????? = 0 ???? ????,???? + ???? ????,???? ???? = 0′ ???????? ???????? o Suppose ???? ????,???? exists such that???????? (????,???? = ???? ????,???? and ???????? ????,???? = ???? ????,???? and such that ???? ????,???? = ???? defines differentiable function ???? = ???? ???? o ???????? + ???????? ????????= ???? ????,???? + ???? ????,???? ???? ) ′ ???????? ???????? ???????? ???? (???? (,???? ???? ) ) 0 definition of exact differential equation ???????? ∫ ???? (???? (,???? ???? ))= ∫ 0???????? ???????? ???? ????,???? = ???? gives implicit solutions Theorem 2.6.1- Let functions ????,????,????????,???? ????e continuous in rectangular region ????:(????,????) × ????,???? ; then ???????? ???? ????,???? + ???? ????,???? ) = 0 is exact in ???? if???? ???? ????,???? = ????????????,???? ) ???????? o There exists a function ???? ????,???? satisfyin???? ???? ????,???? = ???? ????,???? and ???? ???????????? = ???? ????,???? iff ???? and ???? satisfy ???? ????,???? = ???? ???????????? ) o If ???? ????,???? = ???? ????,???? and ???? ????,???? = ???? ????,???? , then ???? = ???? should equal ???? = ???? under ???? ???? ???? ???????? ???? ???????? certain conditions o Note: not essential that ???? is rectangular, but without holes in interior ???????? −???? Ex. ????????= 2????+????, ???? 0 = ???? 0 o Nonlinear, not Bernoulli, not separable o Write equivalently: (2???? + ????)???????? = −???? ???????? ???? + 2???? + ???? )???????? = 0 ???????? o Look for solution of form ???? ????,???? = ???? Let ???? = ????????= ????: ???????????? = ???? = ???????? + ℎ(????) let ℎ ???? replace ???? because functions of ???? are constant in ???? ∫ Let ???? = ????????= 2???? + ????: ∫ 2???? + ???????????? = ???? = ???? + ???????? + ???? ???? ( ) ???? ???? replaces ????; constant in ???? Must reconcile two versions of ???? ????,???? ; pieces must match up ???? = ???????? + ℎ ???? ) ???? = ???????? + ???? + ????(????) Can see that ℎ ???? = ???? , ???? ???? = 0 2 ∴ ???? ????,???? = ???????? + ???? = ???? o Given ???? 0 = ???? 0 0 ????) + ???? = ???? → ???? = ???? 2 0 2 0 2 0 ???????? + ???? = ???? 0 ???? + ???????? − ???? 0 0 −????± ???? −4 1 (???? 0) ???? = 2 −????± ???? +4???? 2 = 0 take + square root, because taking − would give ???? ≠ − ???? 2 0 −????+ ???? +4???? 2 0 = 2 ???????? Ex. 4???? + 2???? + ???? − 2???? )???????? = 0 Apply the test that???????? must equal ???? for the equation to be exact. If it is exact, find the general solution. o ???? ???? 2, ???? =????1 o ∴ not exact ???????? Ex. 4???? + 2???? + 2???? − 2???? ) = 0 Test to see if the equation is exact. If it is, find the specific solution ???????? given that ???? 0 = 1. o ???? ???? 2, ???? =????2 → exact o ???? ???? 4???? + 2???? ∫ 4???? + 2???????????? = 2???? + 2???????? + ℎ ???? ( ) o ???? = 2???? − 2???? ???? ∫ 2???? − 2???????????? = 2???????? − ???? + ???? ???? ( ) o ???? = 2???????? − ???? + 2???? = ???? o Given ???? 0 = 1: 2 0 1 − 1 + 2 0 ( )2 = ???? 0 − 1 + 0 = ???? −1 = ???? ???? ????,???? = 2???????? − ???? + 2???? = 12 o Solve for explicit solution for ????: ???? − 2???????? − 2???? + 1 = 0 2 2 ???? − 2???????? + −2???? − 1 = 0) 2????± 4???? −4 1 −2???? −1 ) ???? = 2 2????± 4???? +4 2???? +1) = 2 2????±√4???? +8???? +4 = 2 = 2????±√12???? +4 2 2????± 4 3???? +1) = 2 2????±2√3???? +1 = 2 = ???? ± √???? + 1 2 = ???? + √???? + 1 must satisfy ???? 0 = 1 ∴ ???? ???? = ???? + 3√ + 1 o What if we could have ???? 0 = 0? 2???????? − ???? + 2???? = ???? 2 0 0 − 0 + 2 0 ( )2 = ???? ???? = 0 By quadratic formula, ???? = ???? ± √3???? = ???? ± √ ????| | Not a good IVP (no solution), because ???? not differentiable at origin; initial condition is at a point in an interval where it should not be ???????? Ex. ???????? ????????+ ???????? ????????− sin???? ) = 0 ???? ???????? o ???? ???? ???????? ????????) = ???????????? + ???????????? ???????? ???????? ???? o ???? ???? (???????????????? − sin???? = ???? ???????? + ???????????????????? ???????? o ∴ exact o ???? = ∫???????? ???????????????? 1 ???????? = ???????? + ℎ ???? ) ???????? = ???? + ℎ ????) o ???? = ∫???????? ????????− sin???????????? ???????? 1 = ???????? (????) + cos???? + ???? ????) ???????? = ???????? + cos???? + ???? ????) o ???? = ???? + cos???? = ???? ???????????? = ???? − cos???? ???????? = ln ???? − cos????) ???? = ln ????−cos???? ???? o Sometimes cannot get entirely explicit expression; known as transcendental relation- exponentials, trig functions, etc. in same variable such that variable cannot be isolated completely Integrating Factors o Recall: ????, ???? potentially functions of ???? and ???? Equation exact if???????? = ???? and ????,????,???? ???????? c????ntinuous o If equation fails test to be exact, multiplying by factor ???? makes manipulation into exact form possible o What would ???? satisfy? ????????= ???? ???? (???????? ) = ???????? ) ???? ???? ???????????? + ???????? ???? ???? ???? ???? ???????? ???? could be solved, but not easily o Case 1: ???? = ???? ???? If ???? = ???? ???? , then???????? = 0 ???????? ???? ???? ????????+ ???????? ???? ???????? ???????? ???? ???? + ???????????? ???????? ???????? ???? = ???????? ???? ???????? ???? ???????? ???? −???? ???? −???? = ???? ???? ???? ONLY if ???? ????is a function of only ???? ???????? ???? ???? o Case 2: ???? = ???? ???? If ???? = ???? ???? , then???????? = 0 ???????????? + ???????? ???? ???????? ???? ???????? ???? + ???????? = ???????? ???????? ???? ???? ???????? ???????? ???? = ???????? ???? ???????? ???? ???????? ????????−???????? ????????−???? ???? ???????? = ???? ???? ONLY if ???? is a function of only ???? o Intuitively, only 1 of 2 cases will work (or neither, but not for this class) o Ex. 2???????? + 3???? − 4???? ???? = 0 ????????= 2????, ???? ???? 6???? → not exact Try case 1: ???? ???????? ???? 2????−6???? ???? = 3???? −4???? −4???? = 3???? −4???? ∴ ???? ≠ ???? ????) Try case 2: ????????−???????? 6????−2???? ???? = 2???????? 4???? = 2???????? 2 = ???? ∴ ???? = ???? ????) By case 2,????????= ????2 ???????? ???? ????????− ???? = 0 ???????? ???? ∫− ???????? −2 Let integrating factor ???? = ???? = ????−2ln???? = ???? l( ???? )= ???? −2 −2 ???????? −2 2 ???? ????????− ???? (????)???? = 0 −2 ???????? −3 ???? ????????− 2???? ???? = 0 ???? (???????? −2) = 0 ???????? ∫ ???? (???????? −2)= ∫0???????? ???????? ????????−2 = ???? ???? = ???????? 2 function of ???? only, as it should be Multiply original equation by ???? = ???? ???? : Can just use ???? = ???? because ????’s will cancel 2 2 ???? 2???????? + 3???? − 4???? ????′ = 0 2???????? + 3???? ???? − 4???? ???? = 0 ′ ̃ 2 ̃ 2 Check: ????????= 6???????? , ???? ???? 6???????? Find ???? ????,???? such that ????????= ????, ???? =???????? ̃ 3 2 3 ∫ 2???????? ???????? = ???? ???? + ℎ ???? ( ) ∫ 3???? ???? − 4???? ???????? = 3???? ( ???? ) − ???? + ???? ???? ( ) 3 = ???? ???? − ???? + ???? ???? ( ) 2 3 4 ???? ????,???? = ???? ???? − ???? = ???? leave implicit o Ex. ???? = ????2???? + ???? − 1 first order linear → exact 1 − ???? − ????2???? + ???? = 0 2???? ???? = 1 − ???? − ???? , ???? = 1 ???? ???? −1, ???? =????0 → not exact yet Try case 1: ????????−???????? = −1−0 = −1 function of ???? by default; does not contain ???? ???? 1 ???????? By case 1,????????= −???? ???????? ????????+ ???? = 0 Let ???? = ???? 1= ???? ???? ???????? ???????? + ???? ???? = 0 ???? ???????? (???????? ????)= 0 ???????????? ∫ (????????????) = ∫ 0???????? ???????? ???????? = ???? ???? = ???????? −???? ???? does not matter −???? Multiply original equation by ???? ???? = ????: ????−????(1 − ???? − ????2???? + ????′ = 0 ????−???? − ????????−???? − ???? + ???? ????′ −????= 0 −???? ???? −???? ′ −???? (???? − ???? − ???????? )+ ???? ???? = 0 ̃ −???? ̃ −???? Check: ????????= −???? , ???? = −???? Find ???? ????,???? : ∫ ????−????− ???? − ???????? −???????????? = −???? −???? − ???? + ???????? −???? + ℎ ????) −???? −???? ∫ ???? ???????? = ???????? + ???? ????) ???? ????,???? = −???? − ???? −???? + ????????−???? = ???? ????????−???? = ???? + ???? −???? + ???? 2???? ???? ???? = ???? + 1 + ???????? = ???????? + ???? 2????+ 1

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