<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>Differential_Equation_8</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p>Solve \[{@n1@} \frac{d^2y}{dx^2} + {@cc@} y = 0,\ y(0) = {@n4@},\ y^\prime(0) = {@n6@}\] <br> where \({}^\prime\) denotes differentiation with respect to \(x\). <br> y(x) = [[input:ans1]] [[validation:ans1]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>To solve the differential equation 
\[{@n1@} \frac{d^2y}{dx^2} + {@cc@} y = 0\] 
Let \(y = e^{mx}\) and hence \(\frac{dy}{dx}=me^{mx}\) and \(\frac{d^2y}{dx^2} = m^2e^{mx}\).
Substituting for \(y\) gives 
\[{@n1@}m^2e^{mx} + {@cc@}e^{mx} = 0\] 
and dividing throughout by \(e^{mx}\) and {@n1@} gives 
\[m^2 + {@cc/n1@} = 0\] 
which can be solved to give {@m@}.</p> 
<p>
For roots of \(m\) in the form \(a±bi\), the solution to the differential equation is of the form 
\[y = {@e^(ax)@}(Acos(bx)+Bsin(bx))\] 
and therefore 
\[y= Acos({@bvar*x@})+Bsin({@bvar*x@})\] 
Given \(\ y(0) = {@n4@}\) \[ A = {@n4@}\] and \(\ y^\prime(0) = {@n6@}\) 
\[B = {@n6/bvar@}\] 
Therefore 
\[y = {@correctanswer@}\]
</p>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text>2025042500</text>
    </stackversion>
    <questionvariables>
      <text>n1:rand(8)+2;
n2:rand(4)+1;
n2:2*n2-1;
n4:rand(3)+1;
n6:rand(3)+1;
cc:n1*(n2^2);
msol:solve(n1*m^2 + cc = 0,m);
m1:rhs(msol[1]);
m2:rhs(msol[2]);
avar:realpart(m1);
bvar:abs(imagpart(m1));
correctanswer:n4*cos(bvar*x)+(n6/bvar)*sin(bvar*x);
ode:n1 * 'diff(y,x,2) + cc* y;
wronganswer1:a*sin(n2*x) + b*cos(n2*x);
wronganswer2:n4*cos(n2*x);
wronganswer3:sin(n2*x)*n6/n2;
wronganswer4:(n4/2 - %i*n6/(2*n2))*exp(bvar*x*%i)+(n4/2 + %i*n6/(2*n2))*exp(-bvar*x*%i);
wronganswer5:(n4/2 + %i*n6/(2*n2))*exp(bvar*x*%i)+(n4 + %i*n6/(2*n2))*exp(-bvar*x*%i);</text>
    </questionvariables>
    <specificfeedback format="html">
      <text><![CDATA[<p>[[feedback:Result]]</p>]]></text>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\({@n1@} \frac{d^2y}{dx^2} + {@cc@} y = 0,\ y(0) = {@n4@},\ y^\prime(0) = {@n6@}\) gives {@correctanswer@}</text>
    </questionnote>
    <questiondescription format="moodle_auto_format">
      <text/>
    </questiondescription>
    <questionsimplify>1</questionsimplify>
    <assumepositive>0</assumepositive>
    <assumereal>0</assumereal>
    <prtcorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
    </prtpartiallycorrect>
    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
    </prtincorrect>
    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
    <multiplicationsign>dot</multiplicationsign>
    <sqrtsign>1</sqrtsign>
    <complexno>i</complexno>
    <inversetrig>cos-1</inversetrig>
    <logicsymbol>lang</logicsymbol>
    <matrixparens>[</matrixparens>
    <isbroken>0</isbroken>
    <variantsselectionseed/>
    <input>
      <name>ans1</name>
      <type>algebraic</type>
      <tans>correctanswer</tans>
      <boxsize>30</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords/>
      <allowwords/>
      <forbidfloat>1</forbidfloat>
      <requirelowestterms>1</requirelowestterms>
      <checkanswertype>1</checkanswertype>
      <mustverify>1</mustverify>
      <showvalidation>1</showvalidation>
      <options/>
    </input>
    <prt>
      <name>Result</name>
      <value>2.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text>sa1:ev(subst(y=ans1,ode),simp);
sa2:ev(sa1,nouns);
/* Does ans1 satisfy the ODE? */
sa3:fullratsimp(expand(sa2));
/* Does ans1 satisfy y(0)=n4? */
sa4:fullratsimp(expand(ev(ans1,x=0,simp)));
/* Does ans1 satisfy y'(0)=n5? */
sa5:fullratsimp(expand(block([ds],ds:diff(ans1,x),ev(ds,x=0,simp))));
sa6:setify(block(listconstvars:true,listofvars(ans1)));
sa7:elementp(%i,sa6);</text>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans1</sans>
        <tans>correctanswer</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>Result-1-T </trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0</falsescore>
        <falsepenalty/>
        <falsenextnode>1</falsenextnode>
        <falseanswernote>Result-1-F </falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
      <node>
        <name>1</name>
        <description>Does ans1 satisfy the ODE?</description>
        <answertest>AlgEquiv</answertest>
        <sans>sa3</sans>
        <tans>0</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>+</truescoremode>
        <truescore>0.34</truescore>
        <truepenalty/>
        <truenextnode>2</truenextnode>
        <trueanswernote>Result-2-T </trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[<p>Your solution satisfies the ODE.</p>]]></text>
        </truefeedback>
        <falsescoremode>+</falsescoremode>
        <falsescore>0</falsescore>
        <falsepenalty/>
        <falsenextnode>2</falsenextnode>
        <falseanswernote>Result-2-F </falseanswernote>
        <falsefeedback format="html">
          <text><![CDATA[<p>Your solution does not satisfy the ODE.</p>]]></text>
        </falsefeedback>
      </node>
      <node>
        <name>2</name>
        <description>Does ans1 satisfy y(0)=n4?</description>
        <answertest>AlgEquiv</answertest>
        <sans>sa4</sans>
        <tans>n4</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>+</truescoremode>
        <truescore>0.33</truescore>
        <truepenalty/>
        <truenextnode>3</truenextnode>
        <trueanswernote>Result-3-T </trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[<p>Your solution satisfies the initial condition \(y(0) = {@n4@}\).</p>]]></text>
        </truefeedback>
        <falsescoremode>+</falsescoremode>
        <falsescore>0</falsescore>
        <falsepenalty/>
        <falsenextnode>3</falsenextnode>
        <falseanswernote>Result-3-F </falseanswernote>
        <falsefeedback format="html">
          <text><![CDATA[<p>Your solution does not satisfy the initial condition \(y(0) = {@n4@}\).<br />
When we substitute \(x=0\) into {@ans1@} we get \({@sa4@}\neq {@n4@}\).</p>]]></text>
        </falsefeedback>
      </node>
      <node>
        <name>3</name>
        <description>Does ans1 satisfy y'(0)=n5?</description>
        <answertest>AlgEquiv</answertest>
        <sans>sa5</sans>
        <tans>n6</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>+</truescoremode>
        <truescore>0.33</truescore>
        <truepenalty/>
        <truenextnode>4</truenextnode>
        <trueanswernote>Result-4-T </trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[<p>Your solution satisfies the initial condition \(y^{\prime}(0) = {@n6@}\).</p>]]></text>
        </truefeedback>
        <falsescoremode>+</falsescoremode>
        <falsescore>0</falsescore>
        <falsepenalty/>
        <falsenextnode>4</falsenextnode>
        <falseanswernote>Result-4-F </falseanswernote>
        <falsefeedback format="html">
          <text><![CDATA[<p>Your solution does not satisfy the initial condition \(y^{\prime}(0) = {@n6@}\).<br />
When we substitute \(x=0\) into {@ev(diff(ans1,x),simp)@} we get \({@sa5@}\neq {@n6@}\).</p>]]></text>
        </falsefeedback>
      </node>
      <node>
        <name>4</name>
        <description>Check for %i in the studen't answer.</description>
        <answertest>EqualComAss</answertest>
        <sans>ans1</sans>
        <tans>wronganswer4</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>0.8</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>Result-5-T </trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[<p>Please simplify your answer, which should reduce to a real expression.</p>]]></text>
        </truefeedback>
        <falsescoremode>+</falsescoremode>
        <falsescore>0</falsescore>
        <falsepenalty/>
        <falsenextnode>5</falsenextnode>
        <falseanswernote>Result-5-F </falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
      <node>
        <name>5</name>
        <description>Check for a particular wrong (complex) answer.</description>
        <answertest>AlgEquiv</answertest>
        <sans>sa7</sans>
        <tans>true</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>+</truescoremode>
        <truescore>0</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>Result-6-T </trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[<p>Please simplify your answer, which should reduce to a real expression.</p>]]></text>
        </truefeedback>
        <falsescoremode>+</falsescoremode>
        <falsescore>0</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>Result-6-F </falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <deployedseed>849177645</deployedseed>
    <deployedseed>1122123116</deployedseed>
    <deployedseed>1388262788</deployedseed>
    <deployedseed>178794585</deployedseed>
    <deployedseed>1977867665</deployedseed>
    <deployedseed>552621047</deployedseed>
    <deployedseed>639531176</deployedseed>
    <deployedseed>386704012</deployedseed>
    <deployedseed>1857308468</deployedseed>
    <deployedseed>998719148</deployedseed>
    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>correctanswer</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-1-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>2</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>correctanswer-c</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.3300000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-6-F</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>3</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>wronganswer1</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.3400000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-6-F</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>4</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>wronganswer2</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.6700000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-6-F</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>5</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>wronganswer3</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.6700000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-6-F</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>6</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>wronganswer4</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.8000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-5-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>7</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>wronganswer5</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.3400000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-6-T</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
