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  4. Solution of (dy/dx) = (2x - 6y + 7/x - 3y + 4) is

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Q. Solution of $\frac{dy}{dx} = \frac{2x - 6y + 7}{x - 3y + 4} $ is

Differential Equations

Solution:

Put $x - 3y = z$
$\therefore 1-3\frac{dy}{dx} = \frac{dz}{dx}$
$\therefore \frac{1}{3}\left[1-\frac{dz}{dx}\right] = \frac{2z+7}{z+4}$
$\Rightarrow 1-\frac{dz}{dx} = \frac{6z+21}{z+4}$
$\Rightarrow \frac{dz}{dx} = 1- \frac{6z+21}{z+4}$
$= \frac{z+4-6z-21}{z+4} = \frac{-5z-17}{z+4}$
$\therefore \frac{z+4}{5z+17}dz+dx=0$
$\Rightarrow \frac{5z+20}{5z+17}dz+5dx=0$
$\Rightarrow \left(1+\frac{3}{5z+17}\right)dz+5dx=0$
$\Rightarrow z+\frac{3}{5}log \left(5z+17\right)+5x=c$
$\Rightarrow x-3y+\frac{3}{5} log \left(5x - 15y + 117\right) + 5x = C_{1}$
$\Rightarrow 6x - 3y + \frac{3}{5} log \left(5x - 15y + 17\right) = C_{1}$
$\Rightarrow 2x - y + \frac{1}{5} log \left(5x - 15y + 17\right) = C$