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<p>
    <i>This post is part of a series; you can view the next post <a href="[^baseurl]/posts/differential-exam-1.html">here</a>.</i>
</p>
<p>
    Welcome once more to Deadly Boring Math! With the tempestuous wight of Quiz 2 rapidly striding towards us (it's <i>tomorrow</i> in the usual studio time),
    I'm doing some last-minute studying, and figured I'd post some worked solutions here. These are by no means exhaustive; if you don't do your own studying,
    you probably aren't going to pass.
</p>
<h2>WS3.2.1: Rewriting as Matrix Equations</h2>
<p>
    As WS1.2 is trivial and WS2.5 doesn't actually exist, I've skipped over them. The first question in 3.2 is a fairly simple classification task:
    we're given a few systems of linear differential equations, and asked to convert them to matrix form and classify their homogeneity and autonomousity.
</p>
<ul>
    <li>
        `x' = -x + ty, y' = tx - y`. If we let `X = \[x, y]`, and thus `X' = \[x', y']`, we can write this as a matrix multiplication
        `X' = \[\[-1, t], \[t, -1]] X`. Because there isn't a constant vector, this is <i>homogeneous</i>; because the right side of the equation
        has `t` in it, this is <i>non-autonomous</i>.
    </li>
    <li>
        `x' = 3x - z, y' = x + y - z, z' = z + t`. This one is a bit different in that it's 3-dimensional, but the process is still essentially the same.
        Let `X = \[x, y, z]` and `X' = \[x', y', z']`, and this can be written as `X' = \[\[3, 0, -1], \[1, 1, -1], \[0, 0, 1]] X + \[0, 0, t]`.
        This is <i>non-homogeneous</i>, because the constant term is nonzero; it is <i>non-autonomous</i> because `t` appears on the right side.
    </li>
</ul>
<h2>WS3.2.2: Second-Order ODEs to SLDEs</h2>
<p>
    Interestingly enough, when confronted with some second-order ODEs, we can significantly simplify by rewriting as a system of first-order
    linear ODEs. This question asks us to transform the second-order ODE `u'' - 2u' + u = sin(t)` into an SLDE.
</p>
<p>
    The first step is to define some substitutions: `x = u`, `y = u'`, meaning `x' = u'`, and `y' = u''`. Note also that `x' = y`.
    Substituting these values into the equation gives us `y' - 2y + x = sin(t)`: because we have the constraint `x' = y`, this is
    a system of equations. We do some algebra to get `y' = 2y - x + sin(t), x' = y`, which can be written in matrix form as `X' = \[\[0, 1], \[-1, 2]] X + \[0, sin(t)]`.
</p>
<h2>WS3.3.1</h2>
<p>
    This entire worksheet is about solving SLDEs from their matrix forms. We're given the equation `X' = \[\[1, 1], \[4, -2]] X`,
    and we need to find a general solution. The first step is, of course, to find the eigenvalues and eigenvectors: we can use
    the characteristic polynomial method: knowing that the trace is `-1` and the determinant is `-6`, we have `lambda^2 + lambda - 6 = 0`; the roots
    are `2` and `-3`. We solve for each eigenvalue by solving the equation `\[\[1 - lambda, 1], \[4, -2 - lambda]] v = 0`: for `lambda = 2`, this is
    `v = \[1, 1]`, and for `lambda = -3`, this is `v = \[-1, 4]`. These are linearly independent, so we can immediately construct a general solution:
    `X = c_1 e^{-3t} \[-1, 4] + c_2 e^{2t} \[1, 1]`.
</p>
<h2>SA23</h2>
<p>
    Another instance of solving SLDEs from their matrix form! This time, it's `X' = \[\[4, 6], \[2, 5]] X`. We can find the eigenvalues
    by finding the roots of the characteristic polynomial `lambda^2 - 9lambda + 8` - `1` and `8`. These have the corresponding eigenvectors
    `\[-2, 1]` and `\[3, 2]`. These are linearly independent once again, so we can substitute as usual to get the general solution
    `X = c_1 e^{t} \[-2, 1] + c_2 e^{8t} \[3, 2]` 
</p>
<h2>Phase Portraits</h2>
<p>
    I'm not going to go into phase portraits here, but they will probably be on the test. They are, quite simply, a 2d extension of phase lines:
    having solved a system of two linear differential equations, you can plot a bunch of different curves for sensible constant values, and
    note the direction of the derivative along them at various intervals. I highly recommend studying these closely!
</p>
<h2>Final Notes</h2>
<p>
    The quiz is tomorrow in your normal studio room and time. Don't be late! I will, of course, be wearing a hat covered in balloons;
    it would be pretty cool to see someone else adopt the trend as well. Good luck!
</p>
[/]
[=title "Differential Quiz 2"]
[=subject "Calculus"]
[=author "Tyler Clarke"]
[=date "2025-6-2"]
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