<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>Differential_Equation_10</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p>Solve \[{@n1@} \frac{d^2y}{dx^2} - {@bc@} \frac{dy}{dx} + {@cc@} y = 0,\ y(0) = {@n4@},\ y^\prime(0) = {@n6@}\] 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 \[{@n1@} \frac{d^2y}{dx^2} - {@bc@} \frac{dy}{dx} + {@cc@} y = 0\] Let \(y = e^{mx}\) and hence \(\frac{dy}{dx}=me^{mx}\) and \(\frac{d^2y}{dx^2} = m^2e^{mx}\) <br />Substituting for \(y\) gives \[{@n1@}m^2e^{mx} - {@bc@}me^{mx} + {@cc@}e^{mx} = 0\] and dividing throughout by \(e^{mx}\) and {@n1@} gives \[m^2 - {@bc/n1@}m + {@cc/n1@} = 0\] which can be solved to give \(m = {@m1@}\) <br />The solution to the differential equation is then of the form \(y = A{@e^(m1*x)@}+Bx{@e^(m2*x)@}\) <br /><br /> Given \(\ y(0) = {@n4@}\) \[ A = {@n4@}\] and \(\ y^\prime(0) = {@n6@}\) \[{@m1*A@} + B = {@n6@}\] Solving for A and B gives\[ {@answers[1]@} \mbox{ and } {@answers[2]@}\] Therefore \[y = {@correctanswer@}\]</p>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text/>
    </stackversion>
    <questionvariables>
      <text>n1:rand(8)+2;
 n2:rand(11)+1;
 n4:rand(3)+1;
 n6:rand(3)+1;
 bc:n1*2*n2;
 cc:n1*n2*n2;
 
 msol:solve(n1*m^2 - bc*m + cc = 0,m);
 m1:rhs(msol[1]);
 m2:m1;
 answerform:A*e^(m1*x) + B*x*e^(m2*x);
 equ1:diff(answerform,x);
 equ2:subst(0,x,equ1);
 equ3:subst(0,x,answerform);
 answers:linsolve([equ2=n6,equ3 = n4],[A,B]);
 Aans:answers[1];
 Bans:answers[2];
 correctanswer:rhs(answers[1])*e^(m1*x) + rhs(answers[2])*x*e^(m2*x);
 
 wronganswer1:rhs(answers[1])*e^(m1*x);
 wronganswer2:rhs(answers[2])*e^(m2*x);
 wronganswer3:rhs(answers[1])*exp((n2-1)*x) + rhs(answers[2])*e^(m2*x);
 ode:n1 * 'diff(y,x,2)- bc * 'diff(y,x) + cc * y;</text>
    </questionvariables>
    <specificfeedback format="html">
      <text><![CDATA[<p>[[feedback:Result]]</p>]]></text>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\({@n1@} \frac{d^2y}{dx^2} - {@bc@} \frac{dy}{dx} + {@cc@} y = 0,\quad y(0) = {@n4@},\ y^\prime(0) = {@n6@}\) gives \(y = {@correctanswer@}\)</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>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text>sa1:ev(subst(y=ans1,ode),simp);
 sa2:ev(sa1,nouns);
 sa3:fullratsimp(expand(sa2));
 sa4:ev(ans1,x=0);
 sa5:block([ds],ds:diff(ans1,x),ev(ds,x=0));</text>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans1</sans>
        <tans>correctanswer</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>Result-0-T </trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>+</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>1</falsenextnode>
        <falseanswernote>Result-0-F </falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
      <node>
        <name>1</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>sa3</sans>
        <tans>0</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>+</truescoremode>
        <truescore>0.3400000</truescore>
        <truepenalty/>
        <truenextnode>2</truenextnode>
        <trueanswernote>Result-1-T </trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[<p>Your solution satisfies the ODE.</p>]]></text>
        </truefeedback>
        <falsescoremode>-</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>2</falsenextnode>
        <falseanswernote>Result-1-F </falseanswernote>
        <falsefeedback format="html">
          <text><![CDATA[<p>Your solution does not satisfy the ODE.</p>]]></text>
        </falsefeedback>
      </node>
      <node>
        <name>2</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>sa4</sans>
        <tans>n4</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>+</truescoremode>
        <truescore>0.3300000</truescore>
        <truepenalty/>
        <truenextnode>3</truenextnode>
        <trueanswernote>Result-2-T </trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[<p>Your solution satisfies the initial condition \(y(0) = {@n4@}\).</p>]]></text>
        </truefeedback>
        <falsescoremode>-</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>3</falsenextnode>
        <falseanswernote>Result-2-F </falseanswernote>
        <falsefeedback format="html">
          <text><![CDATA[<p>Your solution does not satisfy the initial condition \(y(0) = {@n4@}\).</p>]]></text>
        </falsefeedback>
      </node>
      <node>
        <name>3</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>sa5</sans>
        <tans>n6</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>+</truescoremode>
        <truescore>0.3300000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>Result-3-T </trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[<p>Your solution satisfies the initial condition \(y^{\prime}(0) = {@n6@}\).</p>]]></text>
        </truefeedback>
        <falsescoremode>-</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>Result-3-F </falseanswernote>
        <falsefeedback format="html">
          <text><![CDATA[<p>Your solution does not satisfy the initial condition \(y^{\prime}(0) = {@n6@}\).</p>]]></text>
        </falsefeedback>
      </node>
    </prt>
    <deployedseed>423241456</deployedseed>
    <deployedseed>1715547650</deployedseed>
    <deployedseed>1824904386</deployedseed>
    <deployedseed>1985644496</deployedseed>
    <deployedseed>844134477</deployedseed>
    <deployedseed>686295286</deployedseed>
    <deployedseed>1553156112</deployedseed>
    <deployedseed>221676518</deployedseed>
    <deployedseed>320331731</deployedseed>
    <deployedseed>1365285477</deployedseed>
    <deployedseed>239878302</deployedseed>
    <deployedseed>1269459738</deployedseed>
    <deployedseed>1032014768</deployedseed>
    <deployedseed>789899213</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-0-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-3-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>3</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>wronganswer1</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.6700000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-3-F</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>4</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>wronganswer2</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.3400000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-3-F</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>5</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>wronganswer3</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-3-F</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>6</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>answerform</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.3400000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-3-F</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
