Pergunta

Solve the equation. (dx)/(dt)=(2)/(xe^t+4x) Begin-by separating the variables Choose the correct answer below. A. xe^4xdx=(2)/(e^t)dt B xdx=(2)/(e^t+4x)dt c (e^t)/(2)dx=(1)/(xe^4x)dt D. The equation is already separated.
Solução

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ThiagoElite · Tutor por 8 anos
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To solve the equation \(\frac{dx}{dt} = \frac{2}{xe^{t+4x}}\) by separating the variables, we need to rearrange the terms so that all \(x\)-terms are on one side and all \(t\)-terms are on the other side.<br /><br />Let's start with the given equation:<br />\[<br />\frac{dx}{dt} = \frac{2}{xe^{t+4x}}<br />\]<br /><br />First, multiply both sides by \(xe^{t+4x}\) to get rid of the denominator:<br />\[<br />xe^{t+4x} \cdot \frac{dx}{dt} = 2<br />\]<br /><br />Next, we want to separate the variables \(x\) and \(t\). Notice that \(xe^{t+4x}\) can be written as \(x \cdot e^{t} \cdot e^{4x}\). So, we can rewrite the equation as:<br />\[<br />xe^{4x} \cdot \frac{dx}{dt} = \frac{2}{e^t} \cdot dt<br />\]<br /><br />This matches option A:<br />\[<br />xe^{4x}dx = \frac{2}{e^t}dt<br />\]<br /><br />So, the correct answer is:<br />A. \(xe^{4x}dx = \frac{2}{e^t}dt\)
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