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April 18, 2014

April 18, 2014

Posted by **COFFEE** on Tuesday, July 10, 2007 at 8:05pm.

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Q" + 24 Q' + 200 Q = 12 sin (10t)

r^2 + 24r + 200 = 0

r = -12 +/- 2sqrt(14)*i

Qc(t) = e^-12*(c1*cos(2sqrt(14)t)+c2*sin(2sqrt(14)t))

Qp(t) = A*cos(10t) + B*sin(10t)

Q'p(t) = -10A*sin(10t) + 10B*cos(10t)

Q"p(t) = -100A*cos(10t) - 100B*sin(10t)

(-100A*cos(10t)-100B*sin(10t))+24(-10A*sin(10t)+10B*cos(10t))+200(A*cos(10t)+B*sin(10t))

(100A+240B)*cos(10t)+(100B+240A)*sin(10t)=12*sin(10t)

100A+240B=12 -> 25A + 60B = 3

-240A+100B=0 -> -48A + 20B = 0

B = 36/845, A = 3/169

Q(t)=e^(-12t)(c1*cos(2sqrt(14)t)+c2*sin(2sqrt(14)t))+(3/169)cos(10t)+(36/845)sin(10t)

c1 = -3/169

I(t)=e^(-12t)[(-12*c1+2sqrt(14)*c2)cos(2sqrt(14)t)+(-2sqrt(14)*c1-12*c2)sin(2sqrt(14)t))]

+(10/845)(15*cos(10t)+36*cos(10t))

I(0)=-12*c1+2sqrt(14)*c2+(360/845)=0

c2 = 108/(338sqrt(14))

Q(t)=e^(-12t)((-3/169)cos(2sqrt(14)t)+(108/(338sqrt(14))sin(2sqrt(14))+(1/845)(15cos(10t)+36sin(10t))

Also,

I(t)=e^(-12t)[(144/169)cos(2sqrt(14)t)-(1128/(338sqrt(14))sin(2sqrt(14)t)]

Is this correct? Thanks.

Substitute Q(t) in the differential equation to see if it is correct. The probability of making two mistakes that exactly cancel each other out is pretty small...

Ok, its incorrect. Where am I making a mistake?

Check if Qp satisfies the inhomogeneous equation and if Qc satisfies the homogeneous part.

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