MAT244--2018F > Quiz-4

Q4 TUT 03014

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Victor Ivrii:
Better but still not perfect

http://forum.math.toronto.edu/index.php?topic=1443.0

And what about constants (it should be a general solution)?

Michael Poon:
Right, I think it should be:

$Y(t) = Y_c + Y_p$

$Y(t) = c_1 \cos(3t) + c_2 \sin(3t)+ \sin(3t) \ln| \sec(3t)+ \tan(3t)|-1 + c_3$, where $c_1, c_2, c_3 \in {\rm I\!R}$

Since $c_3$ can absorb the "-1", it becomes:

$Y(t) = c_1 \cos(3t) + c_2 \sin(3t)+ \sin(3t) \ln| \sec(3t)+ \tan(3t)| + c_3$, where $c_1, c_2, c_3 \in {\rm I\!R}$

Victor Ivrii:
$c_3$? There are only 2 const, and $y=1$ is not a solution to homogeneous equation. And to make $\mathbb{R}$ (etc) you type \mathbb{R} !

Yunqi(Yuki) Huang:
For the left-hand side, we could write as $r^2+9=0$.Thus $r=3i$ or $r=-3i.$
Now  $Y_{c}=C_{1}cos3t+C_{2}sin3t$
For the Right-hand side, $Y_{1}(t)=cos3t$$. Y_{2}(t)+sin3t$$. g(t)=9sec^2(3t)$
$$W(t)=\left[ \begin{matrix} cos3t & sin3t \\ -3sin3t& 3cos3t \end{matrix} \right] \tag{3}=3$$
$$W_{1}(t)=\left[ \begin{matrix} 0 & sin3t \\ 1 & 3cos3t \end{matrix} \right] \tag{3}=-sin3t$$
$$W_{2}(t)=\left[ \begin{matrix} cos3t & 0 \\ -3sin3t& 1 \end{matrix} \right] \tag{3}=cos3t$$
We could check the integral table that $\int\tan(t)\sec(t)dt=sec(t)$. $\int\sec(t)dt=ln|sec(t)+tan(t)|$
So, the particular solution is $Y_{p}(t)=Y_{1}\int\frac{g(s)W_{1}(s)}{W(s)}\,ds+Y_{2}\int\frac{g(s)W_{2}(s)}{W(s)}\,ds=-3\cos(3t)\int\tan(3s)\sec(3s)ds+3\sin(3t)\int\sec(3t)ds=-3\cos3t*\frac{1}{3}sec(3t)+3\sin(3t)ln|\sec3t+tan3t|*\frac{1}{3}$
Thus, the general solution is $Y(t)=C_{1}\cos(3t)+C_{2}\sin(3t)+\sin(3t)ln|\sec(3t)+\tan(3t)|-1$