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y" + 27y + 162y = 8 sin (et)

(1) Let C₁ and C₂ be arbitrary constants. The general solution to the related homogeneous differential equation y" +27y + 162y = 0 is the function y₁(x) = C₁ y₁(x) + C'₁ y₂(x) = C₁+C₂[

NOTE: The order in which you enter the answers is important; that is, C₁f(x) + Cag(x) ‡ C₁g(x) + C₂f(x).

(2) The particular solution y(a) to the differential equation

y" + 27y + 162y = sin(e)

is of the form y(x) = y₁ (x) u₁(x) + y2(x) u₂(x) where u₁(x) = and u₂(x) =

(3) It follows that u₁(x) = - and u₂(x) = thus yp(x) =

(4) The most general solution to the non-homogeneous differential equation y" + 27y + 162y = sin(e) is

y = C₁+₂+

Fig: 1