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# MATH-M343/S343 Section 3.5 Notes MATH-S343

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This 6 page Class Notes was uploaded by Kathryn Brinser on Thursday October 13, 2016. The Class Notes belongs to MATH-S343 at Indiana University taught by Michael Jolly in Fall 2016. Since its upload, it has received 6 views. For similar materials see Honors Differential Equations in Mathematics at Indiana University.

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Date Created: 10/13/16

S343 Section 3.5 Notes- Nonhomogeneous Equations; Method of Undetermined Coefficients 10-13-16 ′′ ′ Nonhomogeneous equation- ???? ???? = ???? + ???? ???? ???? + ???? ???? ???? = ???? ???? where ????,????,???? are given continuous functions on open interval ???? o ???? ???? = 0 gives homogeneous equation ???? + ???? ???? ???? + ???? ???? ???? = 0( ) Theorem 3.5.1- Suppose ???? ????1and ???? ???? a2e solutions of the nonhomogeneous ODE such that ???? + ???? ???? ???? + ???? ???? ???? = ???? ???? = ???? or ???? 1 the2 ???? − ???? =2???? ???? 1 ???? ???? = 0 is a solution to the homogeneous ODE o If ????1,????2} is also a fundamental set of solutions for the homogeneous equation, then ????1 1???? + ???? ????2 2= ???? ???? − 1 ???? = 0 2( ) o Proof: ????1,????2satisfy ???? ????1 (???? = ???? ???? and ???? ???? [2](???? = ???? ????( ) Take difference of equations: ???? 1](???? − ???? ????[ 2 (???? = ???? ???? − ???? ???? = 0 Using linearity, ????1???? − ???? ????2= ???? ???? − 1 2] So ???? ????1− ???? 2](???? = 0, which shows ???? −1???? is2solution to homogeneous ODE Because all solutions can be expressed as linear combinations of fundamental sets of solutions (Theorem 3.2.4), we can write ????1− ???? 2 ???? ???? 1 1 ???? =2 2 Theorem 3.5.2- The general solution of the nonhomogeneous ODE can be written as ???? = ???? ???? = ???? ???? ???? + ???? ???? ???? + ???? ???? , where ???? ,???? are a fundamental set of solutions to 1 1 2 2 1 2 corresponding homogeneous ODE, ???? ,???? 1re2arbitrary constants, and ???? is some specific solution of the nonhomogeneous ODE o Proof: ????1− ???? 2 ???? ????1 1???? ???? h2 2s if we identify ???? ???? 1s an arbitrary solution ???? ???? to ) ???? + ???? ???? ???? + ???? ???? ???? = ???? ???? and ???? ???? as 2pecific solution ???? ???? ( ) ???? ???? − ???? ???? = ???? ???? ???? + ???? ???? ???? ( ) 1 1 2 2 ′′ ′ Since ???? arbitrary, 1 1 + ????2 2+ ???? ???? includes all solutions to ???? + ???? ???? ???? + ???? ???? ???? = ???? ???? and ) can be identified as general solution o To solve nonhomogeneous ODE: Find general solution 1 1 ???? + ???? ????2 2for corresponding homogeneous equation; known as complementary solution, denoted ???? ???? ????r ???? ???? ???? ( ) Find some single particular solution ???? ???? of nonhomogeneous equation Form sum of solutions to get ???? ???? = ???? ????1 1+ ???? ???? ????2 2???? ????) ( ) Method of Undetermined Coefficients o Requires initial assumption about form of ???? ???? , but with coefficients unspecified ′′ ′ o Substitute assumed expression into ???? + ???? ???? ???? + ???? ???? ???? = ???? ???? and determine coefficients to satisfy ???? + ???? ???? ???? + ???? ???? ???? = ???? ???? ( ) o Straight-forward after determining form of ???? ????) o Mainly useful for equations in which assuming ???? is simple; used for problems in which homogeneous ODE has constant coefficients and ???? ???? restricted to limited kinds of functions for nonhomogeneous ODE ???? ???? must consist of polynomials, exponentials, sines, or cosines o Ex. Find a particular solution to ???? − 3???? − 4???? = 3???? .2???? ′′ ′ 2???? Want to find ???? such that ???? − 3???? − 4???? = 3???? Since ???? function replicates itself in differentiation, most likely way to get desired result is to assume ???? ???? = ???????? 2????where ???? is undetermined constant ′ 2???? ????′′ = 2???????? 2???? ???? (???? = 4???????? Substitute into ODE: ′′ ′ 2???? ???? − 3???? − 4????2???? 3???? 2???? 2???? o = 4???????? − 6???????? − 4???????? o = −6???????? 2???? −6???????? 2????= 3???? 2???? −1 ???? = 2 −1 2???? Particular solution ???? ???? =2 ???? 1 2???? General solution to ODE:1 1???? + 2 2 − 2 (would find ???? ???? if we wanted to find 1 ,2???? ) o Ex. Find a particular solution to ???? − 3????′ − 4???? = 2sin????. Like example above, assume ???? ???? = ????sin????, where ???? is undetermined constant ???? = ????cos???? ???? = −????sin???? Substitute into ODE: o −????sin???? − 3????cos???? − 4????sin???? = 2sin???? o −5????sin???? − 3????cos???? = 2sin???? o −3????cos???? − 5???? + 2 sin???? = 0 o (5???? + 2 sin???? + 3????cos???? = 0 This must hold for all ????, so it must hold specifically for ???? = 0, 2 So then 5???? + 2 = 0,3???? = 0, leading to contradiction in value of ???? → ???? ???? ≠ ????sin???? Must include cosine term in ????, so now assume ???? ???? = ????sin???? + ????cos???? ′ ???? = ????cos???? − ????sin???? ???? = −????sin???? − ????cos???? Substitute: o (−????sin???? − ????cos???? − 3????cos???? − 3????sin???? − 4????sin???? + 4????cos???? = 2sin???? ) o (−???? + 3???? − 4???? sin???? + −???? − 3???? − 4???? cos???? = 2sin???? o −???? − 3???? − 4???? = 0 −???? + 3???? − 4???? = 2 o −3???? − 5???? = 0 3???? − 5???? = 2 o −3???? = 5???? 3???? − 5( −5????) = 2 3 o ???? = −5???? 3???? + 25???? = 2 3 3 o ???? = −5( ) 34???? = 2 −15 17−5 3 6 3 o ???? = = ???? = = 51 17 −5 3 34 17 Particular solution: ???? ???? = sin???? + cos???? ′′17 ′ 17 2 o Ex. Find a particular solution to ???? − 3???? − 4???? = 4???? − 1. Assume ???? ???? = ???????? + ???????? + ???? (general polynomial of 2 degree) ???? = 2???????? + ???? ′′ ???? = 2???? Substitute: 2???? − 6???????? + 3???? − 4???????? + 4???????? + 4???? = 4???? − 1 2 2 2 −4???????? + −6???? + 4???? ???? + 2???? − 3???? − 4???? = 4???? − 1 −4???????? = 4???? 2 o ???? = −1 −6???? + 4???? = 0 o −6 + 4???? = 0 o 4???? = 6 3 o ???? = 2 2???? − 3???? −94???? = −1 o −2 − − 4???? = −1 −13 2 o 2 − 4???? = −1 −13 o 4???? = 2 + 1 −11 1 −11 o ???? = 2 (4) = 8 2 3 11 Particular solution: ???? ???? = −???? + ???? 2 8 ′′ ???? o Ex. Find a particular solutio???? to ???? − 3???? − 4???? = −8???? cos2????. ???? Assume ???? is product of ???? and linear combination of cos2???? and sin2????; ???? ???? = ???????? cos2???? + ???????? sin2???? ′ ???? ???? ???? ???? ???? = ???????? cos2???? − 2???????? sin2???? + ???????? sin2???? + 2???????? cos2???? o = −2???? + ???? ???? sin2???? + ???? + 2???? ???? cos2????) ???? ???? = ???????? cos2???? − 2???????? sin2???? − 2???????? sin2???? + 4???????? cos2???? + ???????? sin2???? + ) ( ???? ???? ???? ???? 2???????? cos2???? + 2???????? cos2???? − 4???????? sin2???? ) o = −2???? − 2???? + ???? − 4???? ???? sin2???? + ???? − 4???? + 2???? + 2???? ???? cos2???? ) ???? o = −4???? − 3???? ???? sin2???? + −3???? + 4???? ???? cos2???? ) ???? Substitute: (−4???? − 3???? ???? sin2???? + −3???? + 4???? ???? cos2???? − ???? ((−6???? + 3???? ???? sin2???? + ???? ???? ???? ???? (3???? + 6???? ???? cos2???? − ) ((4???????? cos2???? + 4???????? sin2???? )) = −8???? cos2???? = −4???? − 3???? ???? sin2???? + −3???? + 4???? ???? cos2???? + 6???? − 3???? ???? sin2???? − ) ???? ( ) ???? ???? ???? 3???? + 6???? ???? cos2???? − 4???????? cos2???? − 4???????? ????in2???? ???? = −4???? − 3???? + 6???? − 3???? − 4???? ???? sin2???? + −3???? + 4???? − 3???? − 6???? − 4???? ???? cos2???? ) = 2???? − 10???? ???? sin2???? + −10???? − 2???? ???? cos2???? ) ???? 2???? − 10???? = 0 o 2???? = 10???? o ???? = 5???? −10???? − 2???? = −8 o 10 5???? + 2???? = 8 o 52???? = 8 8 2 o ???? = 52 = 13 2 10 o ???? = 5( )13 13 10 ???? 2 ???? Particular solution: ???? ???? = 13 ???? cos2???? + 13???? sin2???? o General idea for determining ???? ???? : ???????? ???????? If ???? ???? = ???? , then ???? ???? = ???????? If ???? ???? = sin ???????? or cos ???????? , then ???? ???? = ????sin ???????? + ???? cos ???????? ( ) ???? ????−1 ???? ????−1 If ???? ???? = ???? ???? + ???? ????−1???? + ⋯+ ???? ????1+ ???? , 0hen ???? ???? = ???? ???? + ???? ????−1 ???? + ⋯+ ???? ???? 1 ???? 0 Same principles extend to case where ???? ???? is product of any number of these Suppose ????(????) is sum of two terms, ie. ???? ???? = ???? ???? + ???? ???? and ???? ,???? are solutions of ′′ ′ ′′ 1 ′ 2 1 2 equations ???????? + ???????? + ???????? = ???? ???? an1 ???????? + ???????? + ???????? = ???? ???? 2( ) Then ???? 1 ???? i2 solution of ???????? + ???????? + ???????? = ???? ???? ( ) o To check, substitute ????1+ ???? 2or ???? and use equations above For an equation whose nonhomogeneous function ???? ???? can be expressed as a sum, we can consider simpler fragments and add results together ′′ ′ 2???? ???? o Ex. Find a particular solution of ???? − 3???? − 4???? = 3???? + 2sin???? − 8???? cos2????. ???? − 3???? − 4???? = 3???? , ???? − 3???? − 4???? = 2sin????, and ???? − 3???? − 4???? = −8???? cos2???? ???? −1 2???? 5 3 Particular solution is sum of those from previous examples: ???? ???? = 2 ???? − 17sin???? + 17cos???? + 2 ???? 10 ???? 13 ???? sin2???? + 13 ???? cos2???? Allows solving of complex equations efficiently ′′ ′ −???? o Ex. Find a particular solution to ???? − 3???? − 4???? = 2???? . As earlier, assume ???? ???? = ???????? −???? ′ −???? ???? = −???????? ???? = ???????? −???? Substitute: −???? −???? −???? −???? o ???????? − −3???????? ) − 4???????? = 2???? o = ???????? −???? + 3???????? −????− 4???????? −???? o = 0 −???? −???? o Not possible for 0 = 2???? ; ???? ???? ≠ ???????? Solve for general solution to homogeneous ODE: ???? − 3???? − 4 = 0 o (???? − 4 ???? + 1 = 0 o ???? = −1,4 ???? = ???? ???? −???? + ???? ????4???? ???? 1 2 Can see that ???? ???? we chose is part of homogeneous solution, so it cannot be solution of nonhomogeneous one Need alternative method; solve equation in different way and use result to guide assumption of correct form of ???? or find simpler equation where difficulty occurs and use its solution to suggest how to solve original equation Look for analogous first order equation: ???? + ???? = 2???? Use method of integrating factors to solve: o Let ???? ???? = ???? ∫1????????= ???? ???? ???????? o ???? ( + ????) = 2???? −????(????)???? ???? ???????? o (???????? ????)= 2 ???????????? ???? o ∫ ????????(???????? ) = ∫ 2???????? ???? o ???????? = 2???? −???????? −???? o ???? = 2???????? + ???????? o ???????? −????is general solution to homogeneous equation ???? + ???? = 0 o 2????????−???? is particular solution to nonhomogeneous equation; notice factor of ???? −???? Going back to original equation, assume ???? ???? = ???????????? o ???? = ???????? −????− ???????????? −???? o ???? = −???????? −???? − ???????? −????− ???????????? −????) −???? −???? = −2???????? + ???????????? o Substitute: (−2???????? −???? + ???????????? −????)− 3???????? −???? − 3???????????? −????) − 4???????????? −???? = 2????−???? −???? −???? −5???????? = 2???? ???? = −2 5 −2 Particular solution: ???? ???? = ????????−???? 5 If assumed form of ???? duplicates a solution of corresponding homogeneous equation, modify assumed form by multiplying it by ???? o Occasionally insufficient to remove all duplicates- if so, multiply by ???? again ′′ ′ o Summary of steps to solve IVP consisting of nonhomogeneous equation ???????? + ???????? + ???????? = ????(????), where ????,????,???? ???? ℝ, and given set of initial conditions Find general solution to corresponding homogeneous equation Make sure ???? ???? fits criteria above (if not, use techniques in 3.6) If ???? ???? = ???? 1 + ???? ???? 2 ⋯+ ???? ???? , f????rm ???? “sub-problems,” each containing one ???? ???? suc???? ) that ???????? + ???????? + ???????? = ???? ???? where 1 ≤ ???? ≤ ???? ????ℎ ???? For ???? sub-problem, assume a particular solution???????? ???? consisting of appropriate exponential/polynomial/sine/cosine function(s) If any duplication in assumed form with solutions of homogeneous equation, multip????y ???? ???? by ???? (or ???? if necessary) Find particular solution????????(????) for each sub-problem; then1???? ???? + ????2???? + ⋯+ ???? ???? ????s particular solution for full nonhomogeneous equation Form sum of general homogeneous solution and particular nonhomogeneous solution to get general nonhomogeneous solution: ???? ???? = ???? ???? 1 1 ???? ???? ???? 2 2 ???? ) ( ) Use initial conditions given to fi1d 2 ,???? ′′ ′ o Table for the Particular Solution of ???????? + ???????? + ???????? = ???????????? ) ???????????? ) ???????????? ) ???? ???? = ???? ???? + ???? ????????−1 + ???? ???? ???? + ???? ????????−1 + ⋯+ ???? ???? + ???? ) ???? ???? ????−1 ???? ????−1 1 0 ⋯+ ???? ????1+ ???? 0 ???????????? ???? ???????? ???? ???? ????????+ ???? ????−1 ????????−1 + ⋯+ ???? ????1+ ???? ???? 0) ???????? ???????? ???? ???? ????−1 ???????? ???????????? ???? sin???????? ???? ???? ????????+ ???? ????−1 ???? + ⋯+ ???? ????1+ ???? ???? 0) sin???????? + ???????????? ???? ???????? cos???????? ???? ???? ????????+ ???? ????−1 ????????−1 + ⋯+ ???? ????1+ ???? ???? 0) ????????cos???????? Note: ???? is smallest nonnegative integer that assures no term in ???? ???? is solution of corresponding ???? homogeneous equation For real and distinct solution to characteristic equation, ???? = number of times 0 is a root For repeated roots, ???? = root For complex roots, ???? + ???????? = root Proof of the Method of Undetermined Coefficients ???? ????−1 o Case 1: ???? ???? = ???????????? = ???? ???? ???? ???? ????−1 ???? + ⋯+ ???? ????1+ ???? 0 ???????? + ???????? + ???????? = ???? ???? +???????? ???? ????−1????????−1 + ⋯+ ???? ???? 1 ???? 0 To get particular solution, assume ???? ???? = ???? ???? + ???? ????????−1 + ⋯+ ???? ???? + ???? ′ ????−1 ????−2 ???? 2 ????−1 1 0 ???? = ???????? ???? ???? + ???? − 1 ????) ????−1???? + ⋯+ 3???? ???? 3 2???? ???? + 2 1 ???? = ???? ???? − 1 ???? ???? ????−2 + ???? − 1 ???? − 2 ???? ) ????????−3 + ⋯+ 6???? ???? + 2???? ???? ????−1 3 2 Substitute into ODE: ???? ???? ???? − 1 ???? ???????? ????−2 + ???? − 1 ???? − 2 ???? ) ????−1????????−3 + ⋯+ 6???? ???? 3 2???? 2 )+ ???? ???????? ???? ????−1 + (???? − 1 ???? ????????−2 + ⋯+ 3???? ???? + 2???? ???? + ???? ) + ???? ???? ???? + ???? ????????−1 + ⋯+ ???? ???? + ???? )= ???? ????−1 ????−1 3 2 1 ???? ????−1 1 0 ???? ???? + ???? ????−1 ???? + ⋯+ ???? ????1+ ???? 0 Equating coefficients with same powers of ???? leads to system of equations o ???????? 0 ???????? +02???????? = ????0 0 ⋮ o ???????? ????−1 + ???????????? ???? ???? ????−1 o ???????? ???? ???? ???? ???? o Given ???? ≠ 0, ????????= ???? , and remaining equations determine rest of1???? ,2 ,…,????????−1 ???? If ???? = 0,???? ≠ 0, then polynomial on left side is of degree ???? − 1; cannot satisfy first equation in substitution step ′′ To be sure ???????? + ????????′ is of degree ????, must choose ???? to be polynomial of degree ???? + 1; we assume ???? ???? = ???? ???? ???? ???? ???? ????−1????????−1 + ⋯+ ???? ????1+ ???? 0) No constant term present, but no need; constants are solutions to homogeneous equation when ???? = 0 Since ???? ≠ 0, ???? = ???????? and other ???? found similarly ???? ???? ????+1) ???? If ???? = 0,???? = 0, assume ???? ???? = ???? ???? ???? ???? ???? ????−1????????−1 + ⋯+ ???? ???? 1 ???? 0) ????????′′ term gives degree ???? Constant and linear terms omitted, since both are solutions to homogeneous ODE o Case 2: ???? ???? = ???? ???? ????) ???????? ′′ ′ ???? ???????? ???????? + ???????? + ???????? = ???? ???? ???? ???? ( ) Assume ???? ???? = ???? ???? ???? where ???? ???? = ???? ????) ????( ) ′ ′ ???????? ???????? ???? = ???? ???? ???? ) + ???? ( ???? ????) ???? = ???? ′(???? ???????????? + ????(???? ???? ????) ????????+ ???? ( ???? ????)) ???????? + ???? ????(???? ????)) ???????? ′′ ???????? ′ ???????? 2 ???????? o = ???? (???? ???? + 2???? ( ???? ????)) + ???? ????(???? ????)) Substitute into ODE: ????(???? ′(???? ???????????? + 2???? ( ???? ????)) ???????? + ???? ????(???? ????)) ????????)+ ???? ( ???? ????) ???????? + ???? ( ???? ????) ????????)+ ???????? ????(???? ???? ????( ))= ???? ????( ) = ????(???? ′(???? ) ????????+ 2???????? + ???? ???? ????(????′( )) ???????? + ???????? + ???????? + ???? ???? ???? ( ( )) ???????? 2 ???? ????−1 If ???????? + ???????? + ???? ≠ 0, we assume that ???? ???? = ???? ???? +???????? ????−1 ???? + ⋯+ ???? ???? 1 ???? 0 Particular solution is of form ???? ???? = ???? ???? ???? + ???? ????−1????????−1 + ⋯+ ???? ???? 1 ???? 0) 2 ???? ????−1 If ???????? + ???????? + ???? = 0 but 2???????? + ???? ≠ 0, must have ???? ???? = ???? ???? ???? +???????? ????−1 ???? + ⋯+ ???? ????1+ ???? 0) Note: when ???????? + ???????? + ???? = 0, ???? ???????? is a solution of homogeneous equation ???? ???? = ???????? ????????(???? ???? + ???? ???? ????−1 + ⋯+ ???? ???? + ???? ) 2 ???? ????−1 1 2 0 ???? ????−1 If ???????? + ???????? + ???? = 0 and 2???????? + ???? = 0, ???? ???? = ???? ???? ???? + ???? ????−1 ???? + ⋯+ ???? ????1+ ???? 0) Implies ???????? and ???????? ????????are homogeneous solutions 2 ???????? ???? ????−1 ???? ???? = ???? ???? (???????????? + ???? ????−1 ???? + ⋯+ ???? ????1+ ???? 0) o Case 3: ???? ???? = ???? ???? ????) ???????? sin???????? or ???? ???? = ???? ???? ???? ) ????????cos???????? Cases very similar; only consider sine here ???????????? −???????????? From Euler’s formula, sin???????? = ???? −???? 2???? ????????+???????? −????(????−???????? ???? Hence ???? ???? = ???? ????????( ) 2???? ???? ???? = ???? ????+???????? ????(???? ???? + ⋯+ ???? ???? + ???? )+ ???? (????−???????? ????(???? ???? + ⋯+ ???? ???? + ???? ) ???????? ???? ???? 1 0 ???????? ???????? 1 0 = ???? (???? ???? + ⋯+ ???? ???? +1???? 0)cos???????? + ???? (???????????? + ⋯+ ???? ???? +1???? 0) sin???????? If ???? + ???????? satisfies characteristic equation corresponding to homogeneous equation, must multiply each polynomial by ???? to increase degree If nonhomogeneous function involves sin???????? and cos????????, usually convenient to treat those terms together, since each may individually result in same form of ????, provided they are not solutions to homogeneous equation

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