What is that ""? It looks like box to me. Is it a special symbol? I'm going to assume it was just a space and ignore it.

Do you mean that your "solution to the associated homogeneous equation" is Yh= 0? If so that is incorrect. Also it should be obvious to you that that equation does not satisfy either of your initial conditions! I suspect what you did was take the solution to the associated homogenous equation to be Yh= C cos(50 t)+ D sin(50 t) and then decide that C and D must to 0 to satisfy the initial conditions. You can't do that. You must find the entire solution first: Also your specific solution Yp is incorrect. It should be clear that since the differential equation has no "odd" derivatives, your solution cannot have a "sin(45 t)" term. Try just X(t)= A cos(45 t).i worked it out as y = 0.03sin45t - 0.036 cos 45t.

Yh = 0 and Yp = 0.03sin45t - 0.036 cos 45t.

can anyone confirm this?

Oh, and there is no "Y" in your problem. Your solution must be X(t), not Y(t).

Again, you cannot determine the coeffients from the initial conditions until2) x'' + 2x' + 26x = 82 cos 4t; x(0) = xa; x'(0) = 0

for xa = 20; 0; 20

somehow i worked it out as Yh = e^-t(cos5t) + (1/5)(e^-t(xa)(sin5t)

Yp = (8/(-8+10i))cos4t + (8/(10+8i))sin4tafteryou have the entire solution. Your particular solution does now involve both cos(4t) and sin(4t) but I have no idea how you could have gotten complex coefficients! I get the coefficient of cos(4x) to be 5 and the coefficient of sin(4x) to be 4!

If you take x(t) (again,notY!) to be A cos(4t)+ B sin(4t) then x'= -4A sin(4t)+ 4B cos(4t) and x"= -16A cos(4t)- 16B sin(4t). Putting those into the differential equation, -16A cos(4t)- 16B sin(4t)- 8A sin(4t)+ 8B cos(4t)+ 26A cos(4t)+ 26B sin(4t)= (-16A+ 8B+ 26A) cos(4t) + (-16B -8A+ 26B) sin(4t)= (10A+ 8B) cos(4t)+ (10B- 8A)sin(4t)= 82 cos(4t). In order for that to be true for all t, we must have 10A+ 8B= 82 and 10B- 8A= 0. I've been know to make arithmetic errors but I dont' believe that can give complex solutions for A and B!

i dun think it's right.

really appreciate it if u guys can help. thanks!