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<title>ODEs you must be able to solve | UWA MATH3022</title>
</head>
<body>
<h1>ODEs you must be able to solve</h1>
<noscript><strong class="noscript">Enable JavaScript for equation rendering to work.</strong></noscript>
<p>
<a href="/">Return to the home page.</a>
</p>
<p>
You <em>must</em> be able to solve the following ODEs.
You <strong>should</strong> be able to <strong>immediately write down</strong> the general solution
in each case:
</p>
<h2 id="decay"><a href="#decay" class="permalink" aria-label="Permalink"></a>First order: exponential decay</h2>
<h3>ODE</h3>
<div class="js-maths important">\frac{\td Y}{\td t} = -\mu Y
</div>
<h3>Solution</h3>
<div class="js-maths">Y (t) = A \ee ^ {-\mu t}
</div>
<h2 id="trigonometric"><a href="#trigonometric" class="permalink" aria-label="Permalink"></a>Second order: trigonometric</h2>
<h3>ODE</h3>
<div class="js-maths important">\frac{\td^2 Y}{{\td x}^2} = -\lambda^2 Y
</div>
<h3>Solution</h3>
<div class="js-maths">Y (x) = A \cos (\lambda x) + B \sin (\lambda x)
</div>
<p>
Note that <span class="js-maths">\cos</span> is an even function and <span class="js-maths">\sin</span> is an odd function:
</p>
<ul>
<li>
If <span class="js-maths">Y</span> is zero at <span class="js-maths">x = 0</span>, you only want <span class="js-maths">\sin</span>
</li>
<li>
If <span class="js-maths">Y</span> has zero slope at <span class="js-maths">x = 0</span>, you only want <span class="js-maths">\cos</span>.
</li>
</ul>
<h2 id="hyperbolic"><a href="#hyperbolic" class="permalink" aria-label="Permalink"></a>Second order: hyperbolic (or exponential)</h2>
<h3>ODE</h3>
<div class="js-maths important">\frac{\td^2 Y}{{\td x}^2} = +\lambda^2 Y
</div>
<h3>Solution</h3>
<div class="js-maths">Y (x) = A \cosh (\lambda x) + B \sinh (\lambda x)
</div>
<p>
OR
</p>
<div class="js-maths">Y (x) = C \ee^ {\lambda x} + D \ee ^ {-\lambda x}
</div>
<p>
When solving boundary value problems,
the first form with the <a href="https://en.wikipedia.org/wiki/Hyperbolic_functions">hyperbolic functions</a> <span class="js-maths">\cosh</span> and <span class="js-maths">\sinh</span>
is much nicer to work with
because <span class="js-maths">\cosh</span> is an even function and <span class="js-maths">\sinh</span> is an odd function:
</p>
<ul>
<li>
If <span class="js-maths">Y</span> is zero at <span class="js-maths">x = 0</span>, you only want <span class="js-maths">\sinh</span>
</li>
<li>
If <span class="js-maths">Y</span> has zero slope at <span class="js-maths">x = 0</span>, you only want <span class="js-maths">\cosh</span>.
</li>
</ul>
<div class="end">
END
</div>
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Last modified: 2020-08-20<br>
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