<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>Differential_Equation_2</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p>Solve the following differential equation: \[ {@sqrt(n1-x^2)@}\frac{dy}{dx}-x(y+{@n2@})=0\] Use the letter c to denote the arbitrary constant occurring in the general solution.<br /> y(x) = [[input:ans1]] [[validation:ans1]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>For the equation \[ {@sqrt(n1-x^2)@}\frac{dy}{dx} -x(y+ {@n2@}) = 0 \] rearranging gives \[\frac{dy}{y+ {@n2@}} = \frac{xdx}{@sqrt(n1-x^2)@}\] To integrate, let \(u = {@n1@}-x^2\) giving \(du = -2dx\) \[\int \frac{dy}{y+ {@n2@}} = \int\frac{-du}{2{@sqrt(u)@}}\]</p>
<p>Hence \[\ln(|y+{@n2@}|) = {@-u^(1/2)@}+c\] and \[y + {@n2@} = {@e^(-u^(1/2)+c)@} = {@e^(c-sqrt(n1-x^2))@}=c{@e^(-(n1-x^2)^(1/2))@}\] <br /> Hence \[y = c{@e^(-(n1-x^2)^(1/2))@}-{@n2@}\]</p>
<p> </p>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text/>
    </stackversion>
    <questionvariables>
      <text>n1:rand(8)+2;
  n2:rand(6)+2;
  correctanswer:c*e^(-(n1-x^2)^(1/2))-n2;
  wronganswer1:Q*e^(-(n1-x^2)^(1/2))-n2;
  wronganswer2:exp(c)*exp(-(n1-x^2)^(1/2))-n2;
  wronganswer3:e^(-(n1-x^2)^(1/2)+c)-n2;
  check2:x/sqrt(n1-x^2);</text>
    </questionvariables>
    <specificfeedback format="html">
      <text><![CDATA[<p>[[feedback:Result]]</p>]]></text>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\({@sqrt(n1-x^2)@}\frac{dy}{dx} -x(y+ {@n2@}) = 0\)  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>
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    <input>
      <name>ans1</name>
      <type>algebraic</type>
      <tans>correctanswer</tans>
      <boxsize>20</boxsize>
      <strictsyntax>1</strictsyntax>
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    </input>
    <prt>
      <name>Result</name>
      <value>3.0000000</value>
      <autosimplify>1</autosimplify>
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      <feedbackvariables>
        <text>studentderivcheck:ev(diff(ans1,x)/(ans1+n2),simp);</text>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans1</sans>
        <tans>correctanswer</tans>
        <testoptions/>
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        <truescore>1.0000000</truescore>
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        <trueanswernote>Result-0-T </trueanswernote>
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          <text/>
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        <falseanswernote>Result-0-F </falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
      <node>
        <name>1</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>studentderivcheck</sans>
        <tans>check2</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>0.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>Result-1-T </trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[<p>Your answer satisfies the differential equation, but is not in the desired simplest format.</p>]]></text>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>Result-1-F </falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
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    <deployedseed>717956073</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+n2</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-1-F</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>3</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>wronganswer1</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-1-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>4</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>wronganswer2</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-1-T</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-1-T</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
