Do you have to solve for the functions

Calculus Level pending

Which of the following systems of linear ODE's is equivalent as the following 3rd order differential equation?

\[ 3 sin\: 3t\: \frac {d^{3} x}{d t^{3}} + 2cos \: 3t\: \frac {d^{2} x}{d t^{2}} - wt\: ln\:t \: \frac{dx}{dt} = e^{xt}\: lnx \]


A.

\( y'_{1} = y_{2} + y_{3} + y'_{2} \)

\( y'_{2} = y_{3} - y_{2} - y'_{1} \)

\( \large y'_{3} = [(e^{y_{1}t} ln \:y_{1}) - (wt ln\: t )\: y_{2}] (1 + \frac{1}{9}cot^{2}\: 3t) - (\frac {2}{3}tan\: 3t) \: y_{3} \)


B.

\( y'_{1} = y_{3}\)

\( y'_{2} = y_{3} - y_{2} - y_{1} \)

\( \large y'_{3} = \frac {(e^{y_{1}t} ln \:y_{1}) + (wt ln\: t )\: y_{2}}{3 sin \: 3t } - (\frac {2}{3}cot\: 3t) \: y_{3} \)


C.

\( y'_{1} = y_{2} \)

\( y'_{2} = y_{3} \)

\( \large y'_{3} = csc^{2} 3t\: [\frac {(e^{y_{1}t} ln \:y_{1}) + (wt ln\: t )\: y_{2}}{3 sin \: 3t }] - (\frac {2}{3}tan\: 3t) \: y_{3} \)


D.

\( y'_{1} = y_{2} \)

\( y'_{2} = y_{3} \)

\( \large y'_{3} = \frac{1}{3} csc\: 3t [(e^{y_{1}t} ln \:y_{1}) + (wt ln\: t )\: y_{2}] - (\frac {2}{3}cot \: 3t) \: y_{3} \)


Note:\( f' (t) = \frac {df}{dt} \)

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