[Solved]Given Several Differential Equations Non Homogeneous Term Periodic Function Equations Foll Q37073872
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Below, you are given several differential equations where the non-homogeneous term is a periodic function. For each of these equations, do the following: (a) Find a few coefficients of the Fourier series of the non-homogeneous term (make sure to use enough terms so that your approximation is reasonable) (b) Use an undetermined-coefficients-style argument to find a few coefficients of the Fourier series for a particular solution of the equation. (c) Plot the particular solution you found, making sure to show several periods Here are the equations: 1) z” + 3x-F(t), where F(t) is the odd function with period 5 such that F(t) 2.5-t for 0 < t < 2.5 2) z” + 10z = G(t), where G(t) is the even function with period 3 such that G(t) = t(1.5-t)for 0 < t < 1.5 Finding particular solutions for an undamped spring-mass sytem under the action of a periodic external force Let us use Fourier series to find an (approximate) particular solution for the differential equation z”(t) +7(t)F), where the force is is a periodic function with period 4 satisfying F(e) 1 0<t<2 First, from last week’s assignment we know how to calculate the Fourier coefficients for the function F, and we also know how to plot it. F[t_] := Piecewise [ { {-1,-2 < t < 0), {1, 0 < t < 2))] Plot [F [Mod [t, 4,-2] ], {t, -6, 6), PlotStyle Blue] maxterms 20; a0- ( 1 / L) * Integrate [F [x] * 1, {x, -L, L a = Table [ ( 1 / L) * Integrate [F [x] * Cos [n * Pi * x / L], {x,-L, L } ] , { n, 1, maxterms} ] b Table [ (1 / L) * Integrate [ F [x] * Sin [ n * Pi * x / L] , {x,-L, L)) , { n, 1, maxterms} ] 0.5 8 2 -0.5 Because the function F is odd, we get that all the a’s are zero as expected, and only some of the b’s are nonzero. If we want to access the values of the b’s, we use expressions of the form b[[n]] b [1]1 bi [211 bi [311 b [4] ] 4 4 0 Since the Fourier series for F only has sines, it makes sense to look for a solution of the form z(t) in(nt/2) Cn Calculating by hand, we can see that then coefficients with those of F(t)- sin(/2) we conclude We can now see what the particular solution looks like, except that we don’t do the infinite sum, we only sum . Equating the given n 27)Or equivalently over the terms we actually calculated c Table[b[ [n]/ (7- n2 Pi 2 /4), [n, 1, maxterms) ] x[t_] := sum [ c [[n11 *Sin [ n * Pi * t / L], {n, 1, maxterns } ] Show transcribed image text Below, you are given several differential equations where the non-homogeneous term is a periodic function. For each of these equations, do the following: (a) Find a few coefficients of the Fourier series of the non-homogeneous term (make sure to use enough terms so that your approximation is reasonable) (b) Use an undetermined-coefficients-style argument to find a few coefficients of the Fourier series for a particular solution of the equation. (c) Plot the particular solution you found, making sure to show several periods Here are the equations: 1) z” + 3x-F(t), where F(t) is the odd function with period 5 such that F(t) 2.5-t for 0
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Answer to Below, you are given several differential equations where the non-homogeneous term is a periodic function. For each of t… . . .
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