MthSc 208, Fall 2011 (Differential Equations)Dr. MacauleyHW 17Due Friday April 14th, 2011(1) Solve the following differential equations:(a)y0=ky(b)y0=-ky(c)y00=k2y(d)y00=-k2y(e)y00+ 3y0+ 2y= 0(f)y00+ 2y0+ 2y= 0(g)y00+ 2y0+y= 0(2) Suppose thatfis a function defined onR(not necessarily periodic).Show that thereis an odd functionfoddand an even functionfevensuch thatf(x) =fodd+feven.Hint: As a guiding example, supposef(x) =eix, and consider cosx=12(eix+e-ix) andisinx=12(eix-e-ix).(3) Express they-intercept off(x) =a02+∞Xn=1ancosnx+bnsinnxin terms of thean’s andbn’s. (Hint: It’snota0ora0/2!)(4) Consider the 2π-periodic functionf(x) =a02+∞Xn=1ancosnx+bnsinnx. Write the Fourierseries for the following functions:(a) The reflection off(x) across they-axis;(b) The reflection off(x) across thex-axis;(c) The reflection off(x) across the origin.(5)(a) The Fourier series of an odd function consists only of sine-terms. What additionalsymmetry conditions onfwill imply that the sine coefficients with even indices willbe zero (i.e., eachb2n= 0)? Give an example of a non-zero function satisfying thisadditional condition.
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