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      <text>19.4.2.3 Task, Find unique solution</text>
    </name>
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      <text><![CDATA[<p>In the above ODE, impose the initial conditions \(x={@x_0@}\) and \(\dfrac{dx}{dt}=0\) at \(t=0\) to find the unique solution.</p>
<p><br></p>
<p>First we can find the values for \(A, B.\)
</p><p>\( A = \) [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p>
<p>\( B = \) [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</p>
<p>Imposing the remaining initial condition we find the solution</p>
<p></p><p>\( x = \) [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</p>]]></text>
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      <text><![CDATA[<p>The solution for the displacement \(x\) is: \[x= A \cos\left( \sqrt{\dfrac{n}{m}} \ t \right) + B \sin \left(\sqrt{\dfrac{n}{m}}\ t\right) \] where \(A, B\) are arbitrary constants.</p>
<p>\[ \dfrac{dx}{dt} = - \sqrt{\dfrac{n}{m}} A \sin \left( \sqrt{\dfrac{n}{m}} \ t \right) + \sqrt{\dfrac{n}{m}} B \cos \left(\sqrt{\dfrac{n}{m}}\ t\right) \]</p>
<p>When \(t=0, \, \dfrac{dx}{dt}=0 \quad \) so that \(\quad \sqrt{\dfrac{n}{m}} \ B=0 \Rightarrow B=0.\)</p>
<p>Therefore \(\quad x=A \cos \left(\sqrt{\dfrac{n}{m}} \ t \right).\)</p>
<p>Imposing the remaining initial condition: when \(t=0,\ x={@x_0@}\) so that \(A= {@x_0@}\) and finally \[x = {@x_0@} \cos \left(\sqrt{\dfrac{n}{m}}\ t\right).\]</p>]]></text>
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      <text>x_0: rand(5)+2;
A: x_0;
B: 0;
ta3: A*cos(sqrt(n/m)*t);</text>
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      <text>\(x = {@ta3@}\)</text>
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