- #1
SqueeSpleen
- 141
- 5
Homework Statement
I was reading a PDE book with a problem of resonance
$$
y_{tt} (x,t) = y_{xx} (x,t) + A \sin( \omega t)
$$
After some work it arrived to a problem of variation of parameters for each odd eigenvalue. To solve it, it uses
$$
y''(t)+a^{2} y(t) = b \sin ( \omega t) \qquad y(0)=0 \quad y'(0)=0
$$
has the solution
$$
y(t) = \dfrac{ b }{ \omega^{2} - a^{2} } \left( \frac{ \omega }{a} \sin (at) - \sin(wt) \right)
$$
I would like to solve this, but as using the method of undetermined coefficients feels like guessing (and for that I can simply verify the solution, which I have already done) I tried to solve it using variation of parameters.
The thing is, I got a real mess, and after a lot of simplifications with trigonometry I arrived to an expression that's closer to that but apparently I still need to do more work.
I wanted to know if there's a more simple way to solve it without using the method of undetermined coefficients or it's really messy if you avoid that.
The equation I arrived to is
$$
y(t) = \dfrac{ b }{ \omega^{2} - a^{2} }
\left( \omega \cos( x \omega) \sin( x \omega) \cos(xa)
+a \sin(x \omega)^{2} \sin (xa)
+a \cos(x \omega) \sin(x \omega) \cos (xa) - \omega \sin (xa) \cos ( x \omega )^{2} \right) + c \sin (x \omega )
$$
Where c has to be determined after putting initial conditions, I think that's ##(-1)( \omega a)##. Anyway, I probably made a mistake in a previous step, and simplying this seems too much work, so I guess there's an easier way to do this.
PD: I have no idea why this doesn't "compile".