<item ident="C3-7223" title="C3 | Homogeneous second-order linear ODE | ver. 7223">
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<presentation>
<material>
<mattextxml>
<div class="exercise-statement">
<p>
<strong>C3.</strong>
</p>
<p>Explain how to find the general solution to each given ODE using exponential functions.</p>
<p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4 \, {x'} = -10 \, {x} - 2 \, {x''}" alt="-4 \, {x'} = -10 \, {x} - 2 \, {x''}" title="-4 \, {x'} = -10 \, {x} - 2 \, {x''}" data-latex="-4 \, {x'} = -10 \, {x} - 2 \, {x''}"/>
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = 243 \, {y} + 54 \, {y'} + 3 \, {y''}" alt="0 = 243 \, {y} + 54 \, {y'} + 3 \, {y''}" title="0 = 243 \, {y} + 54 \, {y'} + 3 \, {y''}" data-latex="0 = 243 \, {y} + 54 \, {y'} + 3 \, {y''}"/>
</p>
</li>
</ol>
</div>
</mattextxml>
<mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>C3.</strong>
</p>
<p>Explain how to find the general solution to each given ODE using exponential functions.</p>
<p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p>
<ol type="a">
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-4%20%5C,%20%7Bx'%7D%20=%20-10%20%5C,%20%7Bx%7D%20-%202%20%5C,%20%7Bx''%7D" alt="-4 \, {x'} = -10 \, {x} - 2 \, {x''}" title="-4 \, {x'} = -10 \, {x} - 2 \, {x''}" data-latex="-4 \, {x'} = -10 \, {x} - 2 \, {x''}">
</p>
</li>
<li>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20243%20%5C,%20%7By%7D%20+%2054%20%5C,%20%7By'%7D%20+%203%20%5C,%20%7By''%7D" alt="0 = 243 \, {y} + 54 \, {y'} + 3 \, {y''}" title="0 = 243 \, {y} + 54 \, {y'} + 3 \, {y''}" data-latex="0 = 243 \, {y} + 54 \, {y'} + 3 \, {y''}">
</p>
</li>
</ol>
</div>
</mattext>
</material>
<response_str ident="response1" rcardinality="Single">
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<response_label ident="answer1" rshuffle="No"/>
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<itemfeedback ident="general_fb">
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<div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = c_{1} e^{\left(\left(2 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 1\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(2 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 1\right) \, t\right)}"/>
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{t}" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{t}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{t}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{t}"/>
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" alt="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}"/>
</p>
</div>
</mattextxml>
<mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20+%201%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20-%201%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(2 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 1\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 1\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i + 1\right) \, t\right)} + c_{2} e^{\left(-\left(2 i - 1\right) \, t\right)}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7Bt%7D" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{t}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{t}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{t}">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" title="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}" data-latex="{y} = k_{1} t e^{\left(-9 \, t\right)} + k_{2} e^{\left(-9 \, t\right)}">
</p>
</div>
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</item>
