Trig derivatives problem

[Andromeda]

Find constants A and B such that the function y = Asin(x) + Bcos(x) satisfies the differential equation y" + y' - 2y = sin(x)

I can find the derivatives just fine, but I'm not sure where to go in order to solve for A or B.

[Martingale]

Find constants A and B such that the function y = Asin(x) + Bcos(x) satisfies the differential equation y" + y' - 2y = sin(x)

I can find the derivatives just fine, but I'm not sure where to go in order to solve for A or B.

plug your derivatives into the equation and solve for A and B. If you want more help show what you have so far.

[Andromeda]

y'(x) = Acosx - Bsinx
y"(x) = -Asinx - Bcosx

(-Asinx - Bcosx) + (Acosx - Bsinx) - 2(Asinx + Bcosx) = sinx

From here I'm not sure how to solve for A or B since there are 2 unknowns and only 1 equation

[Martingale]

y'(x) = Acosx - Bsinx
y"(x) = -Asinx - bcosx

(-Asinx - Bcosx) + (Acosx - Bsinx) - 2(Asinx + Bcosx) = sinx

From here I'm not sure how to solve for A or B since there are 2 unknowns and only 1 equation

Isolate the sines and cosines

you should get something like




or another way to look at it



then

where are functions A and B.

solve this system of two equations and 2 unknowns