Q59 / 160 · AP EAMCET 2020 (22 Sep Shift 2) · Mathematics

Express $\frac{d t}{d x}=\frac{t}{\left(x+t e^{-2 x / 4}\right)}$ in the form of $\frac{d x}{d t}=\phi\left(\frac{x}{t}\right)$

  • Correct answer
    A
    $\frac{x}{t}+e^{-2\left(\frac{x}{t}\right)}$
  • B
    $\frac{x}{t}-e^{-2\left(\frac{x}{t}\right)}$
  • C
    $\frac{x}{t}+e^{2\left(\frac{x}{t}\right)}$
  • D
    $\frac{x}{t}-e^{2\left(\frac{x}{t}\right)}$

Solution

$\frac{d t}{d x}=\frac{t}{x+t e^{\frac{-2 x}{t}}}$ On reciprocal, we get $$ \frac{d x}{d t}=\frac{x+t e^{\frac{-2 x}{t}}}{t}=\left(\frac{x}{t}\right)+e^{-2\left(\frac{x}{t}\right)} $$ Hence, option (1) is correct.

Hence, the correct answer is option: 'A'.

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