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<h1 class="center">Differential Equations Solution Guide</h1><br>
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<div class="def">
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<p>A <a href="differential-equations.html">Differential Equation</a> is
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an equation with a <a href="../sets/function.html">function</a> and
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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" height="123" width="188"><br>
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Example: an equation with the function <b>y</b> and its derivative <b> <span class="intbl"> <em>dy</em> <strong>dx</strong> </span></b> </p>
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</div><br>
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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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<p>Real world examples where
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Differential Equations are used include population growth, electrodynamics, heat
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flow, planetary movement, economic systems and much more!</p>
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<h2>Solving</h2>
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<p>A Differential Equation can be a very natural way of describing something.</p>
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<div class="example">
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<h3>Example: Population Growth</h3>
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<p>This short equation says that a population "N" increases (at any instant) as 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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</div>
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<p>But it is not very useful as it is.</p>
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<p>We need to
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<b>solve</b> it!</p>
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<p>We <b>solve</b> it when we discover <b>the function</b> <b>y</b> (or
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set of functions y) that satisfies the equation, and then it can be used successfully.</p>
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<div class="example">
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<h3>Example: continued</h3>
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<p>Our example is <b>solved</b> with this equation:</p>
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<p class="center larger">N(t) = N<sub>0</sub>e<sup>rt</sup></p>
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<p>What does it say? Let's use it to see:</p>
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<p>With <b>t</b> in months, a population that starts at 1000 (<b>N<sub>0</sub></b>) and a growth rate of 10% per month (<b>r</b>) we get:</p>
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<ul>
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<li>N(1 month) = 1000e<sup>0.1x1</sup> = <b>1105</b></li>
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<li>N(6 months) = 1000e<sup>0.1x6</sup> = <b>1822</b></li>
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<li>etc</li>
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</ul>
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</div>
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<p> </p>
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<p>There is <b>no magic way to solve</b> all Differential Equations.</p>
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<p class="history">But over the millennia great minds have been building on each others work and have discovered different methods (possibly long and complicated methods!) of solving <b>some</b> types of Differential Equations.</p>
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<p id="top-of-page">So let’s take a
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look at some different <b>types of Differential Equations</b> and how to solve them:</p>
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<h2 id="separation">Separation of Variables</h2>
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<div class="center80">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/diff-eq-sep-var-a.svg" alt="Separation of Variables" height="152" width="129"></p>
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<p><a href="separation-variables.html">Separation of Variables</a> can be used when:</p>
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<ul>
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<li>All the y terms (including dy) can be moved to one side
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of the equation, and</li>
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<li>All the x terms (including dx) to the other side.</li>
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</ul>
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<p>If that is the case, we can then integrate and simplify to get the the
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solution.</p>
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<div style="clear:both"></div>
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</div>
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<h2 id="first-order">First Order Linear</h2>
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<div class="center80">
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<p><a href="differential-equations-first-order-linear.html">First Order Linear Differential Equations</a> are of this type:</p>
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<div class="center large">
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<span class="intbl"><em>dy</em><strong>dx</strong></span> + P(x)y = Q(x)
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</div>
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<div><a href="#homogeneous"><br>
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</a></div>Where <b>P(x)</b> and <b>Q(x)</b> are functions of x.
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<p>They are "First Order" when there is only <b> <span class="intbl"> <em>dy</em> <strong>dx</strong> </span></b> (not <b> <span class="intbl"> <em>d<sup>2</sup>y</em> <strong>dx<sup>2</sup></strong> </span></b> or <b> <span class="intbl"> <em>d<sup>3</sup>y</em> <strong>dx<sup>3</sup></strong> </span></b> , etc.)</p>
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<p>Note: a <b>non-linear</b> differential equation is often hard to solve, but we can sometimes approximate it with a linear differential equation to
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find an easier solution.</p>
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</div>
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<h2 id="homogeneous">Homogeneous Equations</h2>
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<div class="center80">
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<p><a href="differential-equations-homogeneous.html">Homogeneous Differential Equations</a> look like this:</p>
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<div class="center large">
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<span class="intbl"> <em>dy</em> <strong>dx</strong> </span> = F ( <span class="intbl"> <em>y</em> <strong>x</strong></span> )
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</div>
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<div><a href="#bernoulli"><br>
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</a></div>We can solve them by using a change of variables:
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<p class="center large">v = <span class="intbl"> <em>y</em> <strong>x</strong> </span></p>
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<p>which can then be solved using <a href="separation-variables.html">Separation of Variables</a> .</p>
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</div>
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<h2 id="bernoulli">Bernoulli Equation</h2>
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<div class="center80">
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<p><a href="differential-equations-bernoulli.html">Bernoull Equations</a> are of this general form:</p>
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<p class="center"><span class="large"><span class="intbl"><em>dy</em><strong>dx</strong></span> + P(x)y = Q(x)y<sup>n</sup></span><br>
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where n is any Real Number but not 0 or 1</p>
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<ul>
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<li>When n = 0 the equation can be solved as a First Order Linear
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Differential Equation.</li>
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<li>When n = 1 the equation can be solved using Separation of
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Variables.</li>
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</ul> <p>For other values of n we can solve it by substituting <span class="large">u = y<sup>1−n</sup></span> and turning it into a linear differential equation (and then solve that).</p>
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</div>
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<h2 id="second">Second Order Equation</h2>
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<div class="center80">
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<p><a href="differential-equations-second-order.html">Second Order (homogeneous)</a> are of the type:</p>
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<div>
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<div class="center large">
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<span class="intbl"></span><span class="intbl"><em>d<sup>2</sup>y</em><strong>dx</strong></span> + P(x)<span class="intbl"> <em>dy</em> <strong>dx</strong></span> + Q(x)y = 0
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</div> </div>
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<a href="#undetermined"></a>
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<p>Notice there is a second derivative <span class="intbl"><em>d<sup>2</sup>y</em> dx<sup>2</sup></span></p>
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<p>The
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<b> general</b> second order equation looks like this</p>
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<p class="large center"> a(x)<span class="intbl"><em>d<sup>2</sup>y</em> dx<sup>2</sup></span> + b(x)<span class="intbl"><em>dy</em> dx</span> + c(x)y = Q(x)</p>
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<p>There are many distinctive cases among these
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equations.</p>
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<p>They are classified as homogeneous (Q(x)=0), non-homogeneous,
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autonomous, constant coefficients, undetermined coefficients etc.</p>
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<p>For <b>non-homogeneous</b> equations the <b>general
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solution</b> is the sum of:</p>
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<ul>
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<li>the solution to the corresponding homogeneous
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equation, and</li>
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<li>the particular solution of the
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non-homogeneous equation</li></ul>
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</div>
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<h2 id="undetermined">Undetermined Coefficients</h2>
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<div class="center80">
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<p>The
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<a href="differential-equations-undetermined-coefficients.html">Undetermined
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Coefficients</a> method works for a non-homogeneous equation like this:</p>
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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> + P(x)<span class="intbl"><em>dy</em><strong>dx</strong></span> + Q(x)y
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= f(x)</p>
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<p>where f(x) is a <b>polynomial, exponential, sine, cosine or a linear combination of those</b>. (For a more general version see Variation of Parameters below)</p>This method also involves making a <b>guess</b>!
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</div>
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<h2 id="variation">Variation of Parameters</h2>
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<div class="center80">
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<p><a href="differential-equations-variation-parameters.html">Variation
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of Parameters</a> is a little messier but works on a wider range of functions than the previous <b>Undetermined
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Coefficients</b>.</p></div>
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<h2 id="exact">Exact Equations and Integrating Factors</h2>
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<div class="center80">
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<p><a href="differential-equations-exact-factors.html">Exact Equations and Integrating Factors</a> can be used for a first-order differential equation like this:</p>
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<p class="center large">M(x, y)dx + N(x, y)dy = 0</p>
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<p>that must have some special function <span class="large">I(x, y)</span> whose <a href="derivatives-partial.html">partial derivatives</a> can be put in place of M and N like this:</p>
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<div>
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<p class="center large"><span class="intbl"><em>∂I</em><strong>∂x</strong></span>dx + <span class="intbl"><em>∂I</em><strong>∂y</strong></span>dy = 0</p>
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</div>Our job is to find that magical function I(x, y) if it exists.
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</div>
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<h2>Ordinary Differential Equations (ODEs) vs Partial Differential Equations (PDEs)</h2>
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<p>All of the methods so far are known as <b>Ordinary Differential Equations</b> (ODE's).</p>
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<div class="words">
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<p>The term <b>ordinary</b> is used in contrast with the term <i>partial</i> to indicate derivatives with respect to only one independent variable.</p>
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</div>
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<p>Differential Equations with unknown multi-variable functions and their
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partial derivatives are a different type and require separate methods to
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solve them.</p>
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<p>They are called <b>Partial Differential Equations</b> (PDE's), and
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sorry, but we don't have any page on this topic yet.</p>
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<p> </p><br>
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<a href="index.html">Calculus Index</a>
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