MA1506CHAP7 - CHAPTER 7 SYSTEMS OF FIRST-ORDER ODEs 7.1....

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Unformatted text preview: CHAPTER 7 SYSTEMS OF FIRST-ORDER ODEs 7.1. ROMEO AND JULIET We all know that many (most) relationships have their ups and downs. Let’s try to model this fact. Romeo loves Juliet, but Juliet believes in a more subtle approach and finds Romeo’s exces- sive enthusiasm rather repulsive - the more he loves her, the less she likes him. On the other hand, when he loses interest, she fears losing him and begins to see his good side. Romeo is more straightforward: his love for Juliet in- creases when she is warm to him, and decreases 1 when she is cold. Let R ( t ) represent Romeo’s feelings and J ( t ) Juliet’s. We can model the lovers’ feelings by: dR dt = aJ R (0) = α dJ dt =- bR J (0) = β (1) where a , b are positive constants and α and β represent their feelings when they first meet. This is a system of simultaneous first- order ODEs . In this case, the equations are linear, so it is easy to solve them. Put, as a trial , R = Ae λt J = Be λt where λ could turn out to be complex - if so, we will as usual interpret the exponential to mean 2 that we are really dealing with sine and cosine functions. [The final solutions for R and J must be real — the feelings are real, not complex!] So Aλe λt = aBe λt and Bλe λt =- bAe λt . So we get Aλ = aB , Bλ =- bA so ignoring special cases we get λ 2 =- ab < 0. Since e iθ = cos θ + i sin θ , this is telling us that the solution is some combination of cos( √ abt ) and sin( √ abt ) i.e. : R ( t ) = C cos( √ abt ) + D sin( √ abt ) J ( t ) = E cos( √ abt ) + F sin( √ abt ) 3 So R (0) = C ˙ R (0) = √ abD J (0) = E ˙ J (0) = √ abF But from equations (1), we see that R (0) = α ˙ R (0) = aJ (0) = aβ J (0) = β ˙ J (0) =- bR (0) =- bα so we have: C = R (0) = α D = ˙ R (0) √ ab = β r a b E = J (0) = β F = ˙ J (0) √ ab =- α r b a Hence R ( t ) = α cos( √ abt ) + β r a b sin( √ abt ) J ( t ) = β cos( √ abt )- α r b a sin( √ abt ) 4 Suppose (for example) that β = 0 and α > 0. Then R ( t ) = α cos( √ abt ) J ( t ) =- α r b a sin( √ abt ) Then the graphs of R ( t ) and J ( t ) are:-2,5 2,5 5 7,5 10 12,5 15-1,6-1,2-0,8-0,4 0,4 0,8 Actually it is more useful to eliminate t and get a direct relation between R and J . A bit of algebra will convince you that: R 2 R 2 max + J 2 J 2 max = 1 5 which can be sketched in the R- J plane: it is an ellipse . This is just for these particular initial condi- tions. Different initial conditions will result in a smaller or a larger ellipse. The full set of ALL possible love-affairs is represented by an infinite set of concentric ellipses: So this diagram tells 6 us everything there is to know about love. Note that at a particular time t , R ( t ) and J ( t ) have definite values, giving a point ( R ( t ) ,J ( t )) in these pictures. The arrows indicate the di- rection of motion of such a point as time goes by. You can check this against the graphs on page 5 - notice that J becomes negative imme- diately after t = 0 if the initial coordinates are 7 (...
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MA1506CHAP7 - CHAPTER 7 SYSTEMS OF FIRST-ORDER ODEs 7.1....

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