Diff Eq. - swapped coefficients help

[Sparky_]

Homework Statement

In an L-R-C circuit, L = 1, R = 2, C = 0.25, E(t) = 50 cos(t)

Find the steady state solution

Homework Equations

The Attempt at a Solution

$$L \frac {di(t)}{dt} + R \frac {dq(t)}{dt} + \frac {q}{C} = 50cos(t)$$

$$\frac {di(t)}{dt} + 2 \frac {dq(t)}{dt} + 4q = 50cos(t)$$

$$q(t) = e^{-t}(c1cos(sqrt(12)t + c2sin(sqrt(12)t)$$

next annihilate 50*cos(t)
$$(D^2 +1) (D^2 + 2D + 4) = (D^2 +1)50cos(t)$$

$$roots = +/- i$$

$$qp(t) = Acos(t) + Bsin(t)$$

$$[-Acos(t) - Bsin(t)] - 2Asin(t) + 2Bcos(t) + 4Acos(t) + 4Bsin(t) = 50cos(t)$$

$$cos(t)[-A+2B+4A] + sin(t)[-B-2A+4B] = 50cos(t)$$

$$3A+2B = 50$$

$$3B-2A = 0$$

$$A = \frac {150}{13}$$

$$B = \frac {100}{13}$$

$$qp(t) = \frac {150}{13}cos(t) + \frac {100}{13}sin(t)$$

The book gets
$$qp(t) = \frac {100}{13}cos(t) - \frac {150}{13}sin(t)$$

I'm off with swapped coefficients and the sign.

Do I have something crossed?
Where is my error?

Thanks
-Sparky

 

[HallsofIvy]

Homework Statement

In an L-R-C circuit, L = 1, R = 2, C = 0.25, E(t) = 50 cos(t)

Find the steady state solution

Homework Equations

The Attempt at a Solution

$$L \frac {di(t)}{dt} + R \frac {dq(t)}{dt} + \frac {q}{C} = 50cos(t)$$

$$\frac {di(t)}{dt} + 2 \frac {dq(t)}{dt} + 4q = 50cos(t)$$

Are we to assume that i= dq/dt?

$$q(t) = e^{-t}(c1cos(sqrt(12)t + c2sin(sqrt(12)t)$$

How did you get that? If you took i= dq/dt so that di/dt= d²q/dt², then characteristic equation is r²+ 2r+ 4= (r+2)²= 0 and the general solution (to the associated homogeneous equation) is
$$q(tt)= c1e^{-2t}+ c2 te^{-2t}$$

next annihilate 50*cos(t)
$$(D^2 +1) (D^2 + 2D + 4) = (D^2 +1)50cos(t)$$

$$roots = +/- i$$

$$qp(t) = Acos(t) + Bsin(t)$$

$$[-Acos(t) - Bsin(t)] - 2Asin(t) + 2Bcos(t) + 4Acos(t) + 4Bsin(t) = 50cos(t)$$

$$cos(t)[-A+2B+4A] + sin(t)[-B-2A+4B] = 50cos(t)$$

$$3A+2B = 50$$

$$3B-2A = 0$$

$$A = \frac {150}{13}$$

$$B = \frac {100}{13}$$

$$qp(t) = \frac {150}{13}cos(t) + \frac {100}{13}sin(t)$$

The book gets
$$qp(t) = \frac {100}{13}cos(t) - \frac {150}{13}sin(t)$$

I'm off with swapped coefficients and the sign.

Do I have something crossed?
Where is my error?

Thanks
-Sparky

 

[Sparky_]

You are correct on the first (homogeneous solution) - I had an obvious error in my work. I didn't recognize the obvious solution and made a mistake in factoring.

(However, I don't need that solution to get the steady state) I need the particular qp(t)

the $$(D^2 +1) (D^2 + 2D + 4) = (D^2 +1)50cos(t)$$

the second term is the solution to the homogeneous - as you pointed out $$q(tt)= c1e^{-2t}+ c2 te^{-2t}$$

The $$(D^2 +1)$$

$$roots = +/- i$$

with all the equating coefficients is where my error is.

Thoughts?

-Sparky