The solution of $(6 x + 7 y - 4) d x + (7 x - 4 y + 3) d y = 0$ is:

3 x^{2} + 7 x y - 2 y^{2} - 4 x + 3 y = c

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6 x^{2} - 4 y^{2} + 7 x y - 4 x + 3 y = c

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3 x^{2} + 14 x y - 4 y^{2} - 4 x + 3 y = c

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6 x^{2} + 4 x - 4 y^{2} + 4 x + 3 y = c

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Solution

The correct option is A $3 x^{2} + 7 x y - 2 y^{2} - 4 x + 3 y = c$
$\Rightarrow \frac{d y}{d x} + \frac{(6 x + 7 y - 4)}{(7 x - 4 y + 3)} = 0$
put $x = X + h; y = Y + k$
$6 h + 7 k - 4 = 0...........1$
$7 h - 4 k + 3 = 0............2$
Put $Y = v X$
$\Rightarrow v + X \frac{d v}{d X} + \frac{6 + 7 v}{7 - 4 v} = 0$
$\Rightarrow X \frac{d v}{d X} + \frac{6 + 14 v - 4 v^{2}}{7 - 4 v} = 0$
$\int \frac{7 - 4 v}{3 + 7 v - 2 v^{2}} d v + \int 2 \frac{d X}{X} = log c$
$\Rightarrow log X^{2} (3 + 7 v - 2 v^{2}) = log c$
$\Rightarrow 3 X^{2} + 7 X Y - 2 Y^{2} = c$
$\Rightarrow 3 {(x - h)}^{2} + 7 (x - h) (y - k) - 2 {(y - k)}^{2} = c$
$\Rightarrow 3 x^{2} + 7 x y - 2 y^{2} + x (- 6 h - 7 k) y (4 k - 7 h) = c$
$\Rightarrow 3 x^{2} + 7 x y - 2 y^{2} - 4 x + 3 y = c$ $[f r o m 1 a n d 2]$

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