Undetermined Coefficients

[bwat687]

Use the method of undetermined coefficients to find one solution of
y'' + 2y' + 2y = (10t + 7)e^(-t)cos(t) + (11t + 25)e^(-t)sin(t)

Homogeneous solution: y(t) = c1e^(-t)cost + c2e(-t)sint

g1(t) = (10t + 7)e^(-t)cos(t) & g2(t) = (11t + 25)e^(-t)sin(t)

Assumed particular solution (for g2(t)):

Yp(t) = (At^2 + Bt + C)e^(-t)sint

Yp'(t) = (At^2 + Bt + C)(-e^(-t)sint + e^(-t)cost) + (2At + B)(e^(-t)sint)

Yp''(t) = (At^2 + Bt + C)(-2e^(-t)cost) + (2At + B)(2e^(-t)cost - 2e^(-t)sint) + (2Ae^(-t)sint)

Substituting back into the right side of the original equation and simplifying:

t(4A + 2B)(e^(-t)cost) + 2Ae^(-t)sint = (11t + 25)e^(-t)sint

Im not really sure if i am equating the coefficients right would i just do this?....

2Ae^(-t)sint = 25e^(-t)sint --> A = 25/2

t(4A + 2B)e^(-t)cost = 11te^(-t)sint --> 50 + 2B = 0 --> B = -1/25

And since there is no C in the final equation does that just mean its zero for the particular solution or do i have a mistake in my derivatives/simplification calculations?

Thus my Yp(t) for g2(t) would equal ((25/2)t^2 - (1/25)B)(e^(-t)sint)

Can anybody tell me what im doing wrong or if i have any mistakes? And then for the final particular solution would i just do the same thing for g1(t) and then add that to g2(t)......Any help is appreciated thanks!

 

[royhaas]

You have a problem somewhere since you have a cosine term on one side and a sine term on the other side. You can equate coefficients of polynomials, but you can't ignore the part with different trig functions.