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By M. Bocher

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Differentiating this expression and using the fact that Y (t ) is a solution of the equation yields f (t ) = 0. Hence u(t ) + iv(t ) is a complex constant times e −iβt . From this it follows directly that Y (t ) is a linear combination of XRe (t ) and XIm (t ). Note that each of these solutions is a periodic function with period 2π/β. Indeed, the phase portrait shows that all solutions lie on circles centered at the origin. These circles are traversed in the clockwise direction if β > 0, counterclockwise if β < 0.

4. This type of system is called a center. 4 Phase portrait for a center. 3 Repeated Eigenvalues 47 Example. (Spiral Sink, Spiral Source) More generally, consider X = AX where α −β A= β α and α, β = 0. The characteristic equation is now λ2 − 2αλ + α 2 + β 2 , so the eigenvalues are λ = α ± iβ. An eigenvector associated to α + iβ is determined by the equation (α − (α + iβ))x + βy = 0. Thus (1, i) is again an eigenvector. Hence we have complex solutions of the form X (t ) = e (α+iβ)t = e αt 1 i cos βt − sin βt + ie αt sin βt cos βt = XRe (t ) + iXIm (t ).

A) Find a formula involving integrals for the solution of this system. (b) Prove that your formula gives the general solution of this system. 18 Chapter 1 First-Order Equations 10. Consider the differential equation x = x + cos t . (a) Find the general solution of this equation. (b) Prove that there is a unique periodic solution for this equation. (c) Compute the Poincaré map p : {t = 0} → {t = 2π} for this equation and use this to verify again that there is a unique periodic solution. 11. First-order differential equations need not have solutions that are defined for all times.

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