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<quiz>
  <question type="stack">
    <name>
      <text>10.2.5 (ODE)</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p>Find \(a,b,c\) such that \(f(x)=ax^2 + bx + c\) satisfies \[ {@alpha@} x^2f''(x) + {@beta@} xf'(x) + {@gamma@} f(x) = {@p@}\] for all \(x\).<br></p><ul><li>\( a = \) [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</li><li>\( b = \) [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</li><li>\( c = \) [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]<br></li></ul><p><br></p><p><br></p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>If \(f(x) = ax^2 + bx + x\), then \[f'(x) = 2ax + b \quad \text{and} \quad f''(x) = 2a.\] Therefore \begin{align*}<br>{@alpha@} x^2f''(x) + {@beta@} xf'(x) + {@gamma@} f(x) &amp;= {@alpha@} x^2 (2a) + {@beta@} x (2ax + b) + {@gamma@} (ax^2 + bx + c) \\<br>&amp;=({@2*alpha@}a + {@2*beta@}a + {@gamma@} a)x^2 + ({@beta@}b + {@gamma@}b ) x + {@gamma@}c.\end{align*}Setting this equal to \({@p@}\) and equating coefficients we see that: </p><p>\begin{align*}<br>({@2*alpha@}+ {@2*beta@} + {@gamma@}) a &amp;= {@A@},\\<br>({@beta@} + {@gamma@})b &amp;= {@B@},\\<br>{@gamma@}c &amp; = {@C@}.\end{align*}</p><p>Therefore the solutions are \begin{align*}<br> a &amp;=\frac{@A@}{@2*alpha + 2*beta +gamma@} = {@ta1@}, \\<br> b &amp;=\frac{@B@}{@beta + gamma@} = {@ta2@},\ \text{and} \\<br> c &amp; = \frac{@C@}{@gamma@} = {@ta3@}.\end{align*}</p>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0.1</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text>2020081700</text>
    </stackversion>
    <questionvariables>
      <text>a: rand_with_prohib(-3,3,[0])
b: rand(3)+1
c: rand_with_prohib(1,5,[b]);
alpha: rand_with_prohib(-5,5,[0])
beta: rand_with_prohib(2,5,[0,-a])
gamma: rand_with_prohib(2,4,[0,-beta, -2*(a+b)])
A: a*(2*alpha+2*beta+gamma)
B: b*(beta+gamma)
C: c*gamma

f: a*x^2+b*x+c
df: diff(f,x)
ddf: diff(f,x,2)
p: A*x^2+B*x+C

ta1: a;
ta2: b;
ta3: c;</text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\( {@alpha@} x^2f''(x) + {@beta@} xf'(x) + {@gamma@} f(x) = {@p@}\) so a = {@ta1@}, b= {@ta2@}, c = {@ta3@}</text>
    </questionnote>
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      <text/>
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    <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>none</multiplicationsign>
    <sqrtsign>1</sqrtsign>
    <complexno>i</complexno>
    <inversetrig>cos-1</inversetrig>
    <logicsymbol>lang</logicsymbol>
    <matrixparens>[</matrixparens>
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      <type>algebraic</type>
      <tans>ta1</tans>
      <boxsize>5</boxsize>
      <strictsyntax>1</strictsyntax>
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      <description/>
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        <name>ans1</name>
        <value>ta1</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>ta2</value>
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        <expectedpenalty>0.0000000</expectedpenalty>
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    <qtest>
      <testcase>2</testcase>
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        <value>ta1+1</value>
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    </qtest>
  </question>
</quiz>
