Question

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

• A. $2x - y + \frac{1}{5}\log (5x - 15y + 17) = C$
• B. $2x - 6y + \frac{1}{5}\log (5x - 15y + 17) = C$
• C. $2x + y + \frac{1}{5}\log (5x - 15y + 17) = C$
• D. none of these.

Answer 1

Answer: (A)

$2x - y + \frac{1}{5}\log (5x - 15y + 17) = C$

Answer 2

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