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224 lines
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224 lines
8.3 KiB
HTML
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<title>Homogeneous Functions</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Homogeneous Functions</h1>
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<h2>Homogeneous</h2>
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<p>To be <b>Homogeneous</b> a function must pass this test:</p>
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<p class="center larger">f(zx, zy)
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=
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z<sup>n</sup> f(x, y)</p>
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<p>In other words</p>
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<div class="tbl">
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<div class="row">
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<span class="lt"><b>Homogeneous</b> is when we can take a function:</span>
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<span class="rt">f(x, y)</span>
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</div>
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<div class="row">
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<span class="lt">multiply each variable by z:</span>
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<span class="rt">f(zx, zy)</span>
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</div>
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<div class="row">
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<span class="lt"><b>and then</b> can rearrange it to get this:</span>
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<span class="rt">z<sup>n</sup> f(x, y)</span>
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</div>
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</div>
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<p>An example will help:</p>
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<div class="example">
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<h3>Example: x + 3y</h3>
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<div class="tbl">
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<div class="row">
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<span class="lt">Start with:</span>
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<span class="rt">f(x, y) = x + 3y</span>
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</div>
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<div class="row">
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<span class="lt">Multiply each variable by z:</span>
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<span class="rt">f(zx, zy) = zx + 3zy</span>
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</div>
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<div class="row">
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<span class="lt">Let's rearrange it by factoring out z:</span>
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<span class="rt">f(zx, zy) = z(x + 3y)</span>
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</div>
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<div class="row">
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<span class="lt">And <b>x + 3y</b> is <b>f(x, y)</b>:</span>
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<span class="rt">f(zx, zy) = z f(x, y)</span>
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</div>
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<div class="row">
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<span class="lt">Which is what we wanted, with n=1:</span>
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<span class="rt">f(zx, zy) = z<sup>1</sup> f(x, y)</span>
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</div>
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</div>
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<p>Yes, <b>x + 3y</b> is homogeneous!</p>
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</div>
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<p>The value of <b>n</b> is called the degree. So in that example the degree is <b>1</b>.</p>
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<div class="example">
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<h3>Example: 4x<sup>2</sup> + y<sup>2</sup></h3>
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<div class="tbl">
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<div class="row">
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<span class="lt">Start with:</span>
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<span class="rt">f(x, y) = 4x<sup>2</sup> + y<sup>2</sup></span>
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</div>
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<div class="row">
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<span class="lt">Multiply each variable by z:</span>
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<span class="rt">f(zx, zy) = 4(zx)<sup>2</sup> + (zy)<sup>2</sup></span>
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</div>
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<div class="row">
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<span class="lt">Which is:</span>
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<span class="rt">f(zx, zy) = 4z<sup>2</sup>x<sup>2</sup> + z<sup>2</sup>y<sup>2</sup></span>
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</div>
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<div class="row">
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<span class="lt">Factoring out <b>z<sup>2</sup></b>:</span>
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<span class="rt">f(zx, zy) = z<sup>2</sup>(4x<sup>2</sup> + y<sup>2</sup>)</span>
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</div>
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<div class="row">
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<span class="lt">And <b>4x<sup>2</sup> + y<sup>2</sup></b> is <b>f(x, y)</b>:</span>
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<span class="rt">f(zx, zy) = z<sup>2</sup> f(x, y)</span>
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</div>
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</div>
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<p>Yes, <b>4x<sup>2</sup> + y<sup>2</sup></b> is homogeneous.</p>
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<p>And its degree is 2.</p>
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</div>
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<p>How about this one:</p>
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<div class="example">
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<h3>Example: x<sup>3</sup> + y<sup>2</sup></h3>
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<div class="tbl">
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<div class="row">
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<span class="lt">Start with:</span>
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<span class="rt">f(x, y) = x<sup>3</sup> + y<sup>2</sup></span>
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</div>
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<div class="row">
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<span class="lt">Multiply each variable by z:</span>
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<span class="rt">f(zx, zy) = (zx)<sup>3</sup> + (zy)<sup>2</sup></span>
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</div>
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<div class="row">
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<span class="lt">Which is:</span>
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<span class="rt">f(zx, zy) = z<sup>3</sup>x<sup>3</sup> + z<sup>2</sup>y<sup>2</sup></span>
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</div>
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<div class="row">
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<span class="lt">Factoring out <b>z<sup>2</sup></b>:</span>
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<span class="rt">f(zx, zy) = z<sup>2</sup>(zx<sup>3</sup> + y<sup>2</sup>)</span>
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</div>
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<div class="row">
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<span class="lt">But <b>zx<sup>3</sup> + y<sup>2</sup></b> is NOT <b>f(x, y)</b>!</span>
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</div>
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</div>
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<p>So <b>x<sup>3</sup> + y<sup>2</sup></b> is NOT homogeneous.</p>
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<p>And notice that x and y have different powers:
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<span class="rt">x<sup>3</sup></span> vs
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<span class="rt">y<sup>2</sup></span>. For polynomial functions that is often a good test.</p>
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</div>
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<p>But not all functions are polynomials. How about this one:</p>
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<div class="example">
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<h3>Example: the function x cos(y/x)</h3>
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<div class="tbl">
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<div class="row">
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<span class="lt">Start with:</span>
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<span class="rt">f(x, y) = x cos(y/x)</span>
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</div>
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<div class="row">
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<span class="lt">Multiply each variable by z:</span>
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<span class="rt">f(zx, zy) = zx cos(zy/zx)</span>
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</div>
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<div class="row">
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<span class="lt">Which is:</span>
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<span class="rt">f(zx, zy) = zx cos(y/x)</span>
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</div>
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<div class="row">
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<span class="lt">Factoring out z:</span>
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<span class="rt">f(zx, zy) = z(x cos(y/x))</span>
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</div>
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<div class="row">
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<span class="lt">And <b>x cos(y/x)</b> is <b>f(x, y):</b></span>
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<span class="rt">f(zx, zy) = z<sup>1 </sup>f(x, y)</span>
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</div>
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</div>
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<p>So <b>x cos(y/x)</b> is homogeneous, with degree of 1.</p>
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<p>Notice that (y/x) is "safe" because (zy/zx) cancels back to (y/x)</p>
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</div>
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<div class="words">
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<img style="float:right; margin: 0 0 5px 10px;" src="images/glass-milk.jpg" alt="milk"><p>Homogeneous, in English, means "of the same kind"</p>
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<p>For example "Homogenized Milk" has the fatty parts spread evenly through the milk (rather than having milk with a fatty layer on top.)</p>
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</div>
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<p class="large">Homogeneous applies to functions like <b>f(x)</b>, <b>f(x, y, z)</b> etc. It is a general idea.</p>
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<h2>Homogeneous Differential Equations</h2>
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<p>A first order <a href="differential-equations.html">Differential Equation</a> is <b>homogeneous</b> when it can be in this form:</p>
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<p class="center"><img src="images/homogneous-equation.svg" alt="homogeneous equation"></p>
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<div class="def">
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<p>In other words, when it can be like this:</p>
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<p class="center larger">M(x, y) dx + N(x, y) dy = 0</p>
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<p><b>And</b> both <b>M(x, y)</b> and <b>N(x, y)</b> are homogeneous functions of the same degree.</p>
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</div>
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<p>Find out more on <span class="center"><a href="differential-equations-homogeneous.html">Solving Homogeneous Differential Equations</a></span>.</p>
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<p> </p>
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<p> </p>
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<div class="related">
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<a href="index.html">Calculus Index</a>
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