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<Text-field style="Heading 3" italic="false" layout="Heading 3"><Font italic="false">Example 2.7 Laplace Solution for Second Order System with Dirac forcing Function</Font></Text-field>
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<Text-field style="Text" layout="Normal">Solve the initial value discussed in Example 2.5 with a Dirac(t) function as the forcing function.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
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<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">This initial value problem can be solved using the Laplace transform technique.  The procedure presented above for the Laplace transformation techinique can be used for solving this example.  When solving a second order differential equation, the initial conditions for both y and <Equation executable="false" style="Text" input-equation="" display="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">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</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(linalg):with(plots):</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(inttrans):</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=diff(y(t),t$2)+2*zeta/tau*diff(y(t),t)+1/tau^2*y(t)=Dirac(t)/tau^2;</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIvLCgtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkieUdGJDYjSSJ0R0YkLUkiJEdGKTYkRi4iIiMiIiIqKkYyRjNJJXpldGFHRiRGM0kkdGF1R0YkISIiLUYoNiRGK0YuRjNGMyomKUY2RjJGN0YrRjNGMyomLUkmRGlyYWNHNiRGKUkoX3N5c2xpYkdGJEYtRjNGO0Y3">LywoLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkieUc2IjYjSSJ0R0YsRi5GLiIiIioqIiIjRi9JJXpldGFHRixGL0kkdGF1R0YsISIiRihGL0YvKiYpRjNGMUY0RipGL0YvKiYtSSZEaXJhY0c2JEYmSShfc3lzbGliR0YsRi1GL0Y2RjQ=</Equation></Text-field>
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<Text-field style="Text" layout="Normal">Even though the equation is second order there is only one equation to be solved.</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">N:=1;</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkiTkc2IiIiIg==">IiIi</Equation></Text-field>
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<Text-field style="Text" layout="Normal">The initial conditions are entered here.</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">y(0):=1;D(y)(0):=0;</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi1JInlHNiI2IyIiISIiIg==">IiIi</Equation></Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pi0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInlHRik2IyIiIUYt">IiIh</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=laplace(eq,t,s);</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIvLCwqJilJInNHRiQiIiMiIiItSShsYXBsYWNlR0YkNiUtSSJ5R0YkNiNJInRHRiRGMkYpRitGK0YpISIiKixGKkYrSSV6ZXRhR0YkRitJJHRhdUdGJEYzRilGK0YsRitGKyooRipGK0Y1RitGNkYzRjMqJilGNkYqRjNGLEYrRisqJEY5RjM=">LywsKiYpSSJzRzYiIiIjIiIiLUkobGFwbGFjZUdGJzYlLUkieUdGJzYjSSJ0R0YnRjBGJkYpRilGJiEiIiosRihGKUklemV0YUdGJ0YpSSR0YXVHRidGMUYmRilGKkYpRikqKEYoRilGM0YpRjRGMUYxKiYpRjRGKEYxRipGKUYpKiRGN0Yx</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqs:=[eq];</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXFzRzYiNyMvLCwqJilJInNHRiQiIiMiIiItSShsYXBsYWNlR0YkNiUtSSJ5R0YkNiNJInRHRiRGM0YqRixGLEYqISIiKixGK0YsSSV6ZXRhR0YkRixJJHRhdUdGJEY0RipGLEYtRixGLCooRitGLEY2RixGN0Y0RjQqJilGN0YrRjRGLUYsRiwqJEY6RjQ=">NyMvLCwqJilJInNHNiIiIiMiIiItSShsYXBsYWNlRzYiNiUtSSJ5RzYiNiNJInRHNiJJInRHNiJJInNHNiIiIiIiIiJJInNHNiIhIiIqLCIiIyIiIkklemV0YUc2IiIiIkkkdGF1RzYiISIiSSJzRzYiIiIiLUkobGFwbGFjZUc2IjYlLUkieUc2IjYjSSJ0RzYiSSJ0RzYiSSJzRzYiIiIiIiIiKigiIiMiIiJJJXpldGFHNiIiIiJJJHRhdUc2IiEiIiEiIiomKUkkdGF1RzYiIiIjISIiLUkobGFwbGFjZUc2IjYlLUkieUc2IjYjSSJ0RzYiSSJ0RzYiSSJzRzYiIiIiIiIiKiQpSSR0YXVHNiIiIiMhIiI=</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dummyvar:={laplace(y(t),t,s)=u};</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkpZHVtbXl2YXJHNiI8Iy8tSShsYXBsYWNlR0YkNiUtSSJ5R0YkNiNJInRHRiRGLUkic0dGJEkidUdGJA==">PCMvLUkobGFwbGFjZUc2IjYlLUkieUc2IjYjSSJ0RzYiSSJ0RzYiSSJzRzYiSSJ1RzYi</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqs:=subs(dummyvar,eqs);</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXFzRzYiNyMvLCwqJilJInNHRiQiIiMiIiJJInVHRiRGLEYsRiohIiIqLEYrRixJJXpldGFHRiRGLEkkdGF1R0YkRi5GKkYsRi1GLEYsKihGK0YsRjBGLEYxRi5GLiomKUYxRitGLkYtRixGLCokRjRGLg==">NyMvLCwqJilJInNHNiIiIiMiIiJJInVHNiIiIiIiIiJJInNHNiIhIiIqLCIiIyIiIkklemV0YUc2IiIiIkkkdGF1RzYiISIiSSJzRzYiIiIiSSJ1RzYiIiIiIiIiKigiIiMiIiJJJXpldGFHNiIiIiJJJHRhdUc2IiEiIiEiIiomKUkkdGF1RzYiIiIjISIiSSJ1RzYiIiIiIiIiKiQpSSR0YXVHNiIiIiMhIiI=</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">vars:=[u];</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkldmFyc0c2IjcjSSJ1R0Yk">NyNJInVHNiI=</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">PP:=genmatrix(eqs,vars,'b1');</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjUFBHNiItSSdtYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IzcjNyMsKCokKUkic0dGJCIiIyIiIkYyKipGMUYySSV6ZXRhR0YkRjJJJHRhdUdGJCEiIkYwRjJGMiokKUY1RjFGNkYy">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">b:=matrix(N,1):for i to N do b[i,1]:=b1[i]:od:evalm(b);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUknbWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiM3IzcjLChJInNHRiciIiIqKCIiI0YtSSV6ZXRhR0YnRi1JJHRhdUdGJyEiIkYtKiQpRjFGL0YyRi0=">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=evalm(inverse(PP)&amp;*b);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pkkkc29sRzYiLUknbWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiM3IzcjKigsKComKUkic0dGJCIiIyIiIilJJHRhdUdGJEYyRjNGMyoqRjJGM0klemV0YUdGJEYzRjFGM0Y1RjNGM0YzRjMhIiJGNEYzLChGMUYzKihGMkYzRjdGM0Y1RjhGMyokRjRGOEYzRjM=">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">solt:=map(invlaplace,sol,s,t);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pkklc29sdEc2Ii1JJ21hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjNyM3IyomLUkkZXhwR0YnNiMsJCooSSJ0R0YkIiIiSSV6ZXRhR0YkRjRJJHRhdUdGJCEiIkY3RjQsJi1JJWNvc2hHRic2IyooRjNGNCkqJilGNiIiJ0Y0LCYqJClGNSIiI0Y0RjRGNEY3RjQjRjRGREY0KUY2IiIlRjdGNCosLUklc2luaEdGJ0Y7RjRGPUY0LCYqJkY1RjRGNkY0RjRGNEY0RjRGQUY3RkZGN0Y0RjQ=">[encoded data, 4696 characters]</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L17" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">y:=solt[1,1];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkieUc2IiomLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJCooSSJ0R0YkIiIiSSV6ZXRhR0YkRi9JJHRhdUdGJCEiIkYyRi8sJi1JJWNvc2hHRig2IyooRi5GLykqJilGMSIiJ0YvLCYqJClGMCIiI0YvRi9GL0YyRi8jRi9GP0YvKUYxIiIlRjJGLyosLUklc2luaEdGKEY2Ri9GOEYvLCYqJkYwRi9GMUYvRi9GL0YvRi9GPEYyRkFGMkYvRi8=">KiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKihJInRHRigiIiJJJXpldGFHRihGLUkkdGF1R0YoISIiRjBGLSwmLUklY29zaEdGJTYjKihGLEYtKSomKUYvIiInRi0sJiokKUYuIiIjRi1GLUYtRjBGLSNGLUY9Ri0pRi8iIiVGMEYtKiwtSSVzaW5oR0YlRjRGLUY2Ri0sJiomRi5GLUYvRi1GLUYtRi1GLUY6RjBGP0YwRi1GLQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L34" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">The value for \317\204 is entered here for plotting purposes.</Font></Text-field>
</Input>
</Group>
<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">tau:=1;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkdGF1RzYiIiIi">IiIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">y:=eval(simplify(y));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkieUc2IiooLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJComSSJ0R0YkIiIiSSV6ZXRhR0YkRi8hIiJGLywoKiYtSSVjb3NoR0YoNiMqJkYuRi8pLCYqJClGMCIiI0YvRi9GL0YxI0YvRjxGL0YvRjhGL0YvKiYtSSVzaW5oR0YoRjZGL0YwRi9GL0Y/Ri9GL0Y4RjE=">KigtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiZJInRHRigiIiJJJXpldGFHRihGLSEiIkYtLCgqJi1JJWNvc2hHRiU2IyomRixGLSksJiokKUYuIiIjRi1GLUYtRi8jRi1GOkYtRi1GNkYtRi0qJi1JJXNpbmhHRiVGNEYtRi5GLUYtRj1GLUYtRjZGLw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pars:=[1.5,2,2.5,3];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkklcGFyc0c2IjcmJCIjOiEiIiIiIyQiI0RGKCIiJA==">NyYkIiM6ISIiIiIjJCIjREYlIiIk</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L39" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">clr:=[red,green,gold,blue];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkY2xyRzYiNyZJJHJlZEdGJEkmZ3JlZW5HRiRJJWdvbGRHRiRJJWJsdWVHRiQ=">NyZJJHJlZEc2IkkmZ3JlZW5HRiRJJWdvbGRHRiRJJWJsdWVHRiQ=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 1 to 4 do p[i]:=plot(subs(zeta=pars[i],y),t=0..15,color=clr[i]):od:</Text-field>
</Input>
</Group>
<Group labelreference="L22" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display(seq(p[i],i=1..4),thickness=4,axes=boxed,labels=[&quot;t&quot;,&quot;y&quot;],title=&quot;Figure Exp. 2.7.1&quot;);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="400" type="two-dimensional" width="400" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" gridlinevisibility="1" legendvisibility="false">[encoded data, 14084 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pars:=[0.2,0.3,0.4,0.5,0.6,0.75];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkklcGFyc0c2IjcoJCIiIyEiIiQiIiRGKCQiIiVGKCQiIiZGKCQiIidGKCQiI3YhIiM=">NygkIiIjISIiJCIiJEYlJCIiJUYlJCIiJkYlJCIiJ0YlJCIjdiEiIw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L40" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p2:=plot([seq(evalf(Re(subs(zeta=pars[i],y))),i=1..6)],t=0..15,thickness=4):</Text-field>
</Input>
</Group>
<Group labelreference="L41" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display(p2,axes=boxed,labels=[&quot;t&quot;,&quot;y&quot;],title=&quot;Figure Exp. 2.7.2&quot;);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="400" type="two-dimensional" width="400" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" gridlinevisibility="1" legendvisibility="false">[encoded data, 19784 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">subs(zeta=1,y);</Text-field>
</Input>
<Output>
<Text-field style="Error" layout="Error">Error, numeric exception: division by zero</Text-field>
</Output>
</Group>
<Group labelreference="L27" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">y1:=limit(y,zeta=1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjeTFHNiIqJiwmKiYiIiMiIiJJInRHRiRGKUYpRilGKUYpLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKiEiIg==">KiYsJiomIiIjIiIiSSJ0RzYiRiZGJkYmRiZGJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYjRichIiI=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot(y1,t=0..15,thickness=4,axes=boxed,labels=[&quot;t&quot;,&quot;y&quot;],title=&quot;Figure Exp. 2.7.3&quot;);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="300" type="two-dimensional" width="400" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" gridlinevisibility="1" legendvisibility="false">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</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L32" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">We observe that the system oscillates for \316\266&lt;1 and does not oscillate for \316\266\342\211\2451.</Font></Text-field>
</Input>
</Group>
<Group labelreference="L31" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>