## Trig derivatives problem

Limits, differentiation, related rates, integration, trig integrals, etc.
Andromeda
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### Trig derivatives problem

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
Posts: 333
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA
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### Re: Trig derivatives problem

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
Posts: 11
Joined: Tue Oct 12, 2010 1:56 am
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### Re: Trig derivatives problem

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
Posts: 333
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA
Contact:

### Re: Trig derivatives problem

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

$\alpha\cdot\sin(x)+\beta\cdot\cos(x)=\sin(x)$

or another way to look at it

$\alpha\cdot\sin(x)+\beta\cdot\cos(x)=1\cdot\sin(x)+0\cdot\cos(x)$

then $\alpha=1, \beta=0$

where $\alpha,\beta$ are functions A and B.

solve this system of two equations and 2 unknowns