Solve $(4 x + 6 y + 3) d x = (6 x + 9 y + 2) d y$

12 (2 x - 3 y) + 5 log (24 x + 36 y + 13) = k .

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12 (2 x - 3 y) + 5 log (24 x - 36 y + 13) = k .

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12 (2 x - 3 y) + 5 log (24 x - 36 y - 13) = k .

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None of these.

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Solution

The correct option is B $12 (2 x - 3 y) + 5 log (24 x + 36 y + 13) = k .$
Given, $(4 x + 6 y + 3) d x = (6 x + 9 y + 2) d y$
$\frac{d y}{d x} = \frac{2 (2 x + 3 y) + 3}{3 (2 x + 3 y) + 2},$
Put $2 x + 3 y = v \Rightarrow 2 + 3 \frac{d y}{d x} = \frac{d v}{d x}$
$\Rightarrow \frac{d v}{d x} - 2 = 3. \frac{2 v + 3}{3 v + 2} = \frac{6 v + 9}{3 v + 2}$
$\Rightarrow \frac{d v}{d x} = \frac{12 v + 13}{3 v + 2}$
$\Rightarrow \frac{3 v + 2}{12 v + 13} d v = d x \Rightarrow \frac{12 v + 13 - 58}{12 v + 13} d v = 4 d x$
$\Rightarrow d v - 5 \frac{d v}{12 v + 13} = 4 d x$
Integrating we get,
$12 (2 x - 3 y) + 5 log (24 x + 36 y + 13) = k .$

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