(25 points) In this problem you will solve the non-homogeneous differential equation
y''+24y'+128 y=Sin(e⁸ˣ)
(1) Let C1 and C2 be arbitrary constants. The general solution to the related homogeneous differential equation y''+24y'+128 y=0$ is the function yh(x)=C1 y1(x)+C2 y2(x)=C1 e^-8x+C2 e^-16x)
NOTE: The order in which you enter the answers is important; that is, C1 f(x)+C2 g(x) C1 g(x)+C2 f(x).
(2) The particular solution yp(x) to the differential equation
y''+24y'+128 y=Sin(e⁸ˣ)
is of the form yp(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''+24y'+128 y=Sin(e⁸ˣ) is y=C1__________ +C2__________ +_______