# Thread: [SOLVED] Verify solution for 2nd order eqn. by taking real part....

1. ## [SOLVED] Verify solution for 2nd order eqn. by taking real part....

Find the particular solutions of $\displaystyle y_{p}(t)$ for the DEs:
$\displaystyle y''+B^{2}y=cos(wt)$ and
$\displaystyle y''+B^{2}y=sin(wt)$
by taking both the real and imaginary parts of the solution:
$\displaystyle z_{p}=\dfrac{sin((w-B)\dfrac{t}{2})}{(w-B)\dfrac{1}{2}} \dfrac{e^{i(w+B)\dfrac{t}{2}}}{i(w+B)}$
Verify the solution for for:$\displaystyle y''+B^{2}y=cos(wt)$

Just to clarify, all other variables except $\displaystyle i$ are real numbers.

Well, here is what i tried...
I started of by trying to take the Real part of $\displaystyle z_{p}$, as the directions state. I think i can do it fine until i get to the denominator on the right side. Once i take the real part of $\displaystyle i(w+b)$, it makes the demoninator zero, which leaves me stuck.

Any helps would be be GREAT!

2. Take the i out from the denominator by multiplying top and bottom by i to get:

$\displaystyle -i\left(\frac{\sin((w-b)t/2)e^{i(w-b)t/2}}{(w-b)(w+b)}\right)$

Now, when you expand the exponent you'll get an x+iy term which when multiplied by the i gives you a real component.

3. Thanks, I got it figured out now.