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      <text>19.1.2.5 Family of Solutions</text>
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<h4>Unique solutions</h4><br>Consider the simple differential equation <span class="math inline">\[ {dy\over dx}=2x.\]</span>
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<p>On integrating, we obtain the general solution <span class="math inline">\[
y=x^2+C\]</span> where <span class="math inline">\(C\)</span> is an arbitrary constant. As <span class="math inline">\(C\)</span> varies we get different solutions, each of which belongs to the family of solutions. The figure below shows some examples.</p>
<figure class="HELM_figure"><img src="@@PLUGINFILE@@/19-1-fig-2-familygraphs.svg" alt="plot of family of solutions" width="426" height="330" class="img-responsive atto_image_button_text-bottom"></figure>
<p>It can be shown that no two members of this family of graphs ever meet and that through each point in the <span class="math inline">\(x\)</span>-<span class="math inline">\(y\)</span> plane passes one, and only one, of these graphs. Hence if we specify
    the boundary condition <span class="math inline">\(y=2\)</span> when <span class="math inline">\(x=0\)</span>, written <span class="math inline">\(y(0)=2\)</span>, then using <span class="math inline">\(y=x^2+c\)</span>: <span class="math inline">\[ 2=0+C\quad \text{so that}\quad C=2\]</span>    and <span class="math inline">\(y=x^2+2\)</span> is the unique solution.</p>]]></text>
<file name="19-1-fig-2-familygraphs.svg" path="/" encoding="base64">[encoded data, 34592 characters]</file>
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