<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>Differential_Equation_3</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p>Solve the following differential equation: \[ \frac{dy}{dx} - {@n1*n2@} -{@n1@}x^2+{@n2@}y^2+y^2x^2 = 0. \] Use the letter \(c\) to denote the arbitrary constant occuring in the general solution.<br> y(x) = [[input:ans1]] [[validation:ans1]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p><span>Rearranging the differential equation gives \[\frac{dy}{dx}-{@n1@}({@n2@}+x^2)+y^2({@n2@}+x^2)=0\] and therefore \[\frac{dy}{dx}=({@n1@}-y^2)({@n2@}+x^2),\] so that </span><br /><span>\[\frac{dy}{{@n1@}-y^2}=({@n2@}+x^2)dx.\]</span><br /><span>Integrating both sides yields: \[\int\frac{dy}{{@n1@}-y^2}=\int({@n2@}+x^2)dx\]</span><br /><span>To integrate the left hand side, rearrange \[\frac{dy}{{@n1@}-y^2}=\frac{dy}{{@n1@}(1-\frac{y^2}{@n1@})}\] Let \(\frac{y}{@sqrt(n1)@} = \tanh(u)\) \[\frac{dy}{@sqrt(n1)@}=(1-\tanh^2(u))du = (1-\frac{y^2}{@n1@})du\] \[\frac{dy}{(1-\frac{y^2}{@n1@})}={@sqrt(n1)@}du.\]</span><br /><span>Therefore \[\int\frac{dy}{{@n1@}-y^2}=\frac{1}{@n1@}\int\frac{dy}{@1-y^2/n1@}=\frac{1}{@n1@}\int{@sqrt(n1)@}du=\frac{u}{@sqrt(n1)@}.\]</span><br /><span>Integrating the right hand side gives \[\int({@n2@}+x^2)dx={@n2@}x+\frac{x^3}{3}+c.\] Therefore \[\frac{u}{@sqrt(n1)@}={@n2@}x+\frac{x^3}{3}+c,\] so that \[u={@sqrt(n1)@}\left({@n2@}x+\frac{x^3}{3}+c\right)\] and hence \[\tanh(u)=\tanh\left({@sqrt(n1)@}\left({@n2@}x+\frac{x^3}{3}+c\right)\right)\]</span><br /><span>Therefore \[\frac{y}{@sqrt(n1)@}=\tanh\left({@sqrt(n1)@}\left({@n2@}x+\frac{x^3}{3}+c\right)\right)\] and \[y={@correctanswer@}\]</span></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:sqrt(n1)*tanh(sqrt(n1)*(n2*x+x^3/3+c));
correctanswer2:sqrt(n1)*tanh(sqrt(n1)*(n2*x+x^3/3)+c);
wronganswer1:sqrt(n1)*tanh(sqrt(n1)*(n2*x+x^3/3+p));
wronganswer2:sqrt(n1)*tanh(sqrt(n1)*(n2*x+x^3/3));
wronganswer3:rat(exponentialize(correctanswer));
wronganswer4:sqrt(n1)*tanh(sqrt(n1)*(n2*x+x^3/3)+p);</text>
    </questionvariables>
    <specificfeedback format="html">
      <text><![CDATA[<p>[[feedback:Result]]</p>]]></text>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\(\frac{dy}{dx} - {@n1*n2@} -{@n1@}x^2+{@n2@}y^2+y^2x^2 = 0\)  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>
    <isbroken>0</isbroken>
    <variantsselectionseed/>
    <input>
      <name>ans1</name>
      <type>algebraic</type>
      <tans>correctanswer</tans>
      <boxsize>20</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>studentderiv:ev(diff(ans1,x),simp);
studentrhs:n1*(n2 + x^2)*(sech(atanh(ans1/sqrt(n1))))^2;
RH:factor(-(-n1*n2-n1*x^2+n2*y^2+y^2*x^2));
RHA:ev(RH,y=ans1);
stuvars:setify(listofvars(ans1));</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>ans1</sans>
        <tans>correctanswer2</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>+</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>Result-1-T </trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>-</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>2</falsenextnode>
        <falseanswernote>Result-1-F </falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
      <node>
        <name>2</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>studentderiv</sans>
        <tans>RHA</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>+</truescoremode>
        <truescore>0.0000000</truescore>
        <truepenalty/>
        <truenextnode>3</truenextnode>
        <trueanswernote>Result-2-T </trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>-</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>Result-2-F </falseanswernote>
        <falsefeedback format="html">
          <text><![CDATA[<p>Your answer does not satisfy the differential equation.</p>]]></text>
        </falsefeedback>
      </node>
      <node>
        <name>3</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>elementp(c,stuvars)</sans>
        <tans>true</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>+</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>Result-3-T </trueanswernote>
        <truefeedback format="html">
          <text><![CDATA[<p>Your answer does satisfy the differential equation, and does contain a constant. However it is in a strange form. Perhaps you could ask your tutor about how to write this in an equivalent form which is "simpler".</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 answer does satisfy the differential equation but either does not have a constant specified or you did not use \(c\) (small case) as instructed.</p>]]></text>
        </falsefeedback>
      </node>
    </prt>
    <deployedseed>1148576613</deployedseed>
    <deployedseed>1448354838</deployedseed>
    <deployedseed>1247893491</deployedseed>
    <deployedseed>2076280914</deployedseed>
    <deployedseed>1396259465</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>correctanswer2</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-1-T</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-3-F</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-3-F</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>5</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>correctanswer/sqrt(n1)</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-2-F</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>6</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>wronganswer3</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-0-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>7</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>wronganswer4</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-3-F</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
