Given the following differential equation $$y''(t)+4y(t)=\frac{1}{4+\cos{2t}}$$ with $y(0)=1$, $y'(0)=0$, find $y(t)$.
Using the Laplace Transform, I've got that $$Y(s)=\frac{s}{s^2+4}+\frac{F(s)}{s^2+4}$$ where $F$ is the Laplace transform of $\frac{1}{4+\cos{2t}}$.
One can immediately observe how the first term is the image of $\cos{2t}$. But what about the second one?
If I use the inverse Laplace Transform of the product $\cfrac{F(s)}{s^2+4}$, I have to compute the convolution between $\cos{2t}$ and $\cfrac{1}{4+\cos{2t}}$, which is $$\int_0^t \frac{\sin(2t-2u)}{4+\cos(2u)}\,du$$
Now, I could use the fact that $\sin(a-b)=\sin a\cos b-\sin b\cos a$.
I was wondering, do I really have to compute this integral? I could also use the variation of parameters method, but I would run into the same integral.
Best Answer
The integral wouldn't be so bad. The identity gives
$$\sin(2t)\int_0^t \frac{\cos(2u)}{4+\cos(2u)}\:du + \cos(2t)\int_0^t \frac{-\sin(2u)}{4+\cos(2u)}\:du$$
The second integral is easy enough
$$\cos(2t)\int_0^t \frac{-\sin(2u)}{4+\cos(2u)}\:du = \frac{1}{2}\cos(2t)\log(4+\cos(2t))$$
The trick to the second integral is to split it up
$$\sin(2t)\int_0^t \frac{4+\cos(2u)-4}{4+\cos(2u)}\:du = t\sin(2t) - 4\sin(2t)\int_0^t \frac{1}{4+\cos(2u)}\:du$$
Then double angle identity on the last integral
$$\int_0^t \frac{1}{5\cos^2u + 3\sin^2u}\:du = \frac{1}{5}\int_0^t \frac{\sec^2u}{1+\frac{3}{5}\tan^2 u}\:du = \frac{1}{\sqrt{15}}\tan^{-1}\left(\sqrt{\frac{3}{5}}\tan t\right)$$
But since the antiderivative needs to be continuous, we need to add a staircase term to raise up the discontinuous points of the function to appropriately connect them. Then adding up all of the results gets us
$$y(t) = \cos(2t) + t\sin(2t) - \frac{4}{\sqrt{15}}\sin(2t)\tan^{-1}\left(\sqrt{\frac{3}{5}}\tan t\right)$$ $$-\frac{4\pi}{\sqrt{15}}\sin(2t)\left\lfloor \frac{1}{\pi}t+\frac{1}{2}\right\rfloor+\frac{1}{2}\cos(2t)\log(4+\cos(2t))$$
This behavior makes perfect sense since the driving force (inhomogeneous term) is in resonance with the homogeneous solution, leading to the growing amplitude term.