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298 lines
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HTML
298 lines
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HTML
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Differential Equations</h1>
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<p>A <span class="center">Differential Equation is a</span>n equation with a <a href="../sets/function.html">function</a> and one or more of its <a href="derivatives-introduction.html">derivatives</a>:</p>
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<p class="center"><img src="images/diff-eq-1.svg" alt="differential equation y + dy/dx = 5x"><br>
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Example: an equation with the function <b>y</b> and its
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derivative<b>
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<span class="intbl">
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<em>dy</em>
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<strong>dx</strong>
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</span>
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</b> </p>
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<h2>Solving</h2>
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<p>We <b>solve</b> it when we discover <b>the function</b> <b>y</b> (or set of functions y).</p>
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<p>There are many "tricks" to solving Differential Equations (<i>if</i> they can be solved!).</p>
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<p>But first: why?</p>
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<h2>Why Are Differential Equations Useful?</h2>
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<p>In our world things change, and <b>describing how they change</b> often ends up as a Differential Equation:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/rabbits.jpg" alt="rabbits" height="147" width="200"></p>
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<h3>Example: Rabbits!</h3>
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<p>The more rabbits we have the more baby rabbits we get.</p>
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<p>Then those rabbits grow up and have babies too! The population will grow faster and faster.</p>
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<p>The important parts of this are:</p>
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<ul>
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<li>the population <b>N</b> at any time <b>t</b></li>
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<li>the growth rate <b>r</b></li>
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<li>the population's rate of change <b><span class="intbl"><em>dN</em><strong>dt</strong></span></b></li>
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</ul>
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<p>Think of <b><span class="intbl"><em>dN</em><strong>dt</strong></span></b> as "how much the population changes as time changes, for any moment in time".</p>
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<p><br></p>
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<p>Let us imagine the growth rate <b>r</b> is <b>0.01</b> new rabbits per week <b>for every current rabbit.</b></p>
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<p>When the population is <b>1000</b>, the rate of change <b><span class="intbl"><em>dN</em><strong>dt</strong></span></b><span class="center"></span> is then 1000×0.01 = <b>10 new rabbits</b> per week.</p>
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<p>But that is only true at a <b>specific time</b>, and doesn't include that the population is constantly increasing. The bigger the population, the more new rabbits we get!</p>
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<p>When the population is <b>2000</b> we get 2000×0.01 = <b>20 new rabbits</b> per week, etc.</p>
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<p>So it is better to say the rate of change (at any instant) is the growth rate times the population at that instant:</p>
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<p class="center large"><span class="intbl"><em>dN</em><strong>dt</strong></span> = rN</p>
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<p>And that is a <b>Differential Equation</b>, because it has a function <b>N(t)</b> and its derivative.</p>
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<p> </p>
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<p>And how powerful mathematics is! That short equation says "the rate of change of the population over time equals the growth rate times the population".</p>
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</div>
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<p><span class="center">Differential</span> Equations can describe how populations change, how heat moves, how springs vibrate, how radioactive material decays and much more. They are a very natural way to describe many things in the universe.</p>
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<h2>What To Do With Them?</h2>
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<p>On its own<span class="center">, a Differential</span> Equation is a wonderful way to express something, but is hard to use.</p>
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<p>So we try to <b>solve</b> them by turning the <span class="center">Differential</span> Equation into a simpler equation without the differential bits, so we can do calculations, make graphs, predict the future, and so on.</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../money/images/coin-stack-add.jpg" alt="coin stack add" height="140" width="106"></p>
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<h3>Example: Compound Interest</h3>
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<p>Money earns interest. The interest can be calculated at fixed times, such as yearly, monthly, etc. and added to the original amount.</p>
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<p>This is called <a href="../money/compound-interest.html">compound interest</a>.</p>
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<p>But when it is compounded <b>continuously</b> then at any time the interest gets added in proportion to the current value of the loan (or investment).</p>
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<p>And as the loan grows it earns more interest.</p>
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<p>Using <b>t</b> for time, <b>r</b> for the interest rate and <b>V</b> for the current value of the loan:</p>
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<p class="center large"><span class="intbl"><em>dV</em><strong>dt</strong></span> = rV</p>
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<p> </p>
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<p><i>And here is a cool thing: it is the same as the equation we got with the Rabbits! It just has different letters. So mathematics shows us these two things behave the same.</i></p>
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<p> </p>
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<p><b>Solving</b></p>
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<p>The <span class="center">Differential</span> Equation says it well, but is hard to use.</p>
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<p>But don't worry, it can be solved (using a special method called <span class="center"><a href="separation-variables.html">Separation of Variables</a></span>) and results in:</p>
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<p class="center larger">V = Pe<sup>rt</sup></p>
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<p>Where <b>P</b> is the Principal (the original loan), and e is <a href="../numbers/e-eulers-number.html">Euler's Number</a>.</p>
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<p>So a continuously compounded loan of $1,000 for 2 years at an interest rate of 10% becomes:</p>
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<div class="tbl">
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<div class="row"><span class="left">V =</span><span class="right"> 1000 × e<sup>(2×0.1)</sup></span></div>
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<div class="row"><span class="left">V =</span><span class="right"> 1000 × 1.22140...</span></div>
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<div class="row"><span class="left">V =</span><span class="right"> $1,221.40 (to the nearest cent)</span></div>
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</div>
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</div>
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<p>So <span class="center">Differential</span> Equations are great at describing things, but need to be solved to be useful.</p>
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<h2>More Examples of Differential Equations</h2>
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<h3>The Verhulst Equation</h3>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/rabbits.jpg" alt="rabbits" height="147" width="200"></p>
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<h3>Example: Rabbits Again!</h3>
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<p>Remember our growth Differential Equation:</p>
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<p class="center large"><span class="intbl"><em>dN</em><strong>dt</strong></span> = rN</p>
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<div style="clear:both"></div>
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<p>Well, that growth can't go on forever as they will soon run out of available food.</p>
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<p>So let's improve it by including:</p>
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<ul>
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<li>the maximum population that the food can support <b>k</b></li>
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</ul>
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<p>A guy called Verhulst figured it all out and got this Differential Equation:</p>
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<p class="center large"><span class="intbl"><em>dN</em><strong>dt</strong></span> = rN(1−N/k)</p>
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<p class="center"><i><b>The Verhulst Equation</b></i></p>
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</div>
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<h3>Simple Harmonic Motion</h3>
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<p>In Physics, Simple Harmonic Motion is a type of periodic motion where the restoring force is directly proportional to the displacement. An example of this is given by a mass on a spring.</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/spring-mass.svg" alt="spring mass"></p>
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<h3>Example: Spring and Weight</h3>
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<p>A spring gets a weight attached to it:</p>
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<ul>
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<li>the weight gets pulled down due to gravity,</li>
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<li>as the spring stretches its tension increases,</li>
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<li>the weight slows down,</li>
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<li>then the spring's tension pulls it back up,</li>
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<li>then it falls back down, up and down, again and again.</li>
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</ul>
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<p>Describe this with mathematics!</p>
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<p> </p>
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<p><b>The weight</b> is pulled down by gravity, and we know from <a href="../physics/force.html">Newton's Second Law</a> that force equals mass times acceleration:</p>
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<p class="center"><b>F</b> = m<b>a</b></p>
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<p>And <a href="../measure/metric-acceleration.html">acceleration</a> is the <a href="second-derivative.html">second derivative</a> of position with respect to time, so:</p>
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<p class="center large"><b>F</b> = m <span class="intbl"><em>d<sup>2</sup>x</em><strong>dt<sup>2</sup>
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</strong></span></p>
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<p> </p>
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<p><b>The spring</b> pulls it back up based on how stretched it is (<b>k</b> is the spring's stiffness, and <b>x</b> is how stretched it is): <b>F = -kx</b></p>
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<p>The two forces are always equal:</p>
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<p class="center large">m <span class="intbl"><em>d<sup>2</sup>x</em><strong>dt<sup>2</sup>
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</strong></span> = −kx</p>
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<p>We have a differential equation!</p>
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<p>It has a function <b>x(t)</b>, and it's second derivative<b> <span class="intbl">
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<em>d<sup>2</sup>x</em>
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<strong>dt<sup>2</sup></strong>
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</span></b></p>
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<p> </p>
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<p><i>Note: we haven't included "damping" (the slowing down of the bounces due to friction), which is a little more complicated, but you can play with it here (press <b>play</b>):</i></p>
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<iframe src="../physics/ispring.html" scrolling="no" style="width:362px; height:392px; overflow:hidden; margin:auto; display:block; border: none;"></iframe>
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</div><p> </p>
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<p><b>Creating</b> a differential equation is the first major step. But we also need to <b>solve</b> it to discover how, for example, the spring bounces up and down over time.</p>
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<h2>Classify Before Trying To Solve</h2>
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<p>So how do we <b>solve</b> them?</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="../measure/images/walking.jpg" alt="walking" height="200" width="101"></p>
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<p>It isn't always easy!</p>
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<p>Over the years wise people have worked out <b>special methods</b> to solve <b>some types</b> of Differential Equations.</p>
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<p>So we need to know <b>what type</b> of Differential Equation it is first.</p>
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<p><i>It is like travel: different kinds of transport have solved how to get to certain places.</i> <i>Is it near, so we can just walk? Is there a road so we can take a car? Or is it in another galaxy and we just can't get there yet?</i></p>
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<div style="clear:both"></div>
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<p>So let us first <b>classify the Differential Equation</b>.</p>
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<p> </p>
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<h3>Ordinary or Partial</h3>
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<p>The first major grouping is:</p>
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<ul>
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<li>"Ordinary Differential Equations" (ODEs) have <b>a single independent variable</b> (like <b>y</b>)</li>
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<li>"Partial Differential Equations" (PDEs) have two or more independent variables.</li>
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</ul>
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<p>We are learning about <b>Ordinary Differential Equations</b> here!</p>
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<p> </p>
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<h3>Order and Degree</h3>
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<p>Next we work out the Order and the Degree:</p>
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<p class="center"><img src="images/diff-eq-2.svg" alt="differential equation order 2, degree 3"></p>
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<h3>Order</h3>
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<p>The Order is the <b>highest derivative</b> (is it a first derivative? a <a href="second-derivative.html">second derivative</a>? etc):</p>
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<div class="example">
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<h3>Example:</h3>
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<p class="center large"><span class="intbl"><em>dy</em><strong>dx</strong></span> + y<sup>2</sup> = 5x</p>
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<p>It has only the first derivative
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<span class="intbl">
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<em>dy</em>
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<strong>dx</strong>
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</span> , so is "First Order"</p>
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</div>
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<div class="example">
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<h3>Example:</h3>
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<p class="center large"><span class="intbl"><em>d<sup>2</sup>y</em><strong>dx<sup>2</sup></strong></span> + xy = sin(x)</p>
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<p>This has a second derivative
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<span class="intbl">
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<em>d<sup>2</sup>y</em>
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<strong>dx<sup>2</sup></strong>
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</span> , so is "Order 2"</p>
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</div>
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<div class="example">
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<h3>Example:</h3>
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<p class="center large"><span class="intbl"><em>d<sup>3</sup>y</em><strong>dx<sup>3</sup></strong></span> + x<span class="intbl"><em>dy</em><strong>dx</strong></span> + y = e<sup>x</sup></p>
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<p>This has a third derivative
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<span class="intbl">
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<em>d<sup>3</sup>y</em>
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<strong>dx<sup>3</sup></strong>
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</span> which outranks the
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<span class="intbl">
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<em>dy</em>
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<strong>dx</strong>
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</span> , so is "Order 3"</p>
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</div>
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<h3>Degree</h3>
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<p>The <a href="../algebra/degree-expression.html">degree</a> is the exponent of the highest derivative.</p>
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<div class="example">
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<h3>Example:</h3>
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<p class="center large">(<span class="intbl"><em>dy</em><strong>dx</strong></span>)<sup>2</sup> + y = 5x<sup>2</sup></p>
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<p>The highest derivative is just dy/dx, and it has an exponent of 2, so this is "Second Degree"</p>
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<p>In fact it is a <b>First Order Second Degree Ordinary Differential Equation</b></p>
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</div>
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<div class="example">
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<h3>Example:</h3>
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<p class="center large"><span class="intbl"><em>d<sup>3</sup>y</em><strong>dx<sup>3</sup></strong></span> + (<span class="intbl"><em>dy</em><strong>dx</strong></span>)<sup>2</sup> + y = 5x<sup>2</sup></p>
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<p>The highest derivative is d<sup>3</sup>y/dx<sup>3</sup>, but it has no exponent (well actually an exponent of 1 which is not shown), so this is "First Degree".</p>
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<p>(The exponent of 2 on dy/dx does not count, as it is not the highest derivative).</p>
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<p>So it is a <b>Third Order First Degree Ordinary Differential Equation</b></p>
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</div>
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<p> </p>
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<p class="center80">Be careful not to confuse order with degree. Some people use the word order when they mean degree!</p>
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<h3>Linear</h3>
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<p>It is <b>Linear</b> when the variable (and its derivatives) has no exponent or other function put on it.</p>
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<p class="center dotpoint">So <b>no</b> y<sup>2</sup>, y<sup>3</sup>, √y, sin(y), ln(y) etc, <b><br>just plain y</b> (or whatever the variable is)</p>
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<p>More formally a <b>Linear Differential Equation</b> is in the form:</p>
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<p class="center large"><span class="intbl"><em>dy</em><strong>dx</strong></span> + P(x)y = Q(x)</p>
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<h2>Solving</h2>
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<p>OK, we have classified our Differential Equation, the next step is solving.</p>
|
||
<p>And we have a <a href="differential-equations-solution-guide.html">Differential Equations Solution Guide</a> to help you.</p>
|
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<p> </p>
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<div class="questions">
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<script>getQ(9399, 9400, 9401, 9402, 9403, 9404, 9405, 9406, 9407, 9408);</script>
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</div>
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<div class="related">
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<a href="derivatives-introduction.html">Derivatives</a>
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<a href="index.html">Calculus Index</a>
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