The solution of dydx=x-2y+32x-y+5

dydx=x−2y+32x−y+5Comparing with dydx= a1x+b1y+c1a2x+b2y+c2∴ a2+b1=0⇒ (2x−y+5)dy= (x−2y+3)dx⇒ 2xdy−ydy+5dy=xdx−2ydx+3dx⇒ 2(x dy+ydx)−ydy+5dy=x dx+3 dx⇒ 2∫D(xy)−∫y dy+∫5 dy=∫x dx+∫3 dx⇒ 2 xy−y22+5y=x22+3x+c⇒ 4xy−y2+10y=x2+6x+c⇒ x2−4xy+y2+6x−10y=c

The solution of dydx=x-2y+32x-y+5

dydx=x−2y+32x−y+5Comparing with dydx= a1x+b1y+c1a2x+b2y+c2∴ a2+b1=0⇒ (2x−y+5)dy= (x−2y+3)dx⇒ 2xdy−ydy+5dy=xdx−2ydx+3dx⇒ 2(x dy+ydx)−ydy+5dy=x dx+3 dx⇒ 2∫D(xy)−∫y dy+∫5 dy=∫x dx+∫3 dx⇒ 2 xy−y22+5y=x22+3x+c⇒ 4xy−y2+10y=x2+6x+c⇒ x2−4xy+y2+6x−10y=c

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The solution of $\frac{dy}{dx} = \frac{x - 2 y + 3}{2 x - y + 5}$

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$x^{2} - 2 xy + y^{2} + 3 x - 5 y = c$

$x^{2} - 4 xy + y^{2} + 6 x - 10 y = c$

$x^{2} - 4 xy + y^{2} + 6 x + 10 y = c$

$x^{2} - 2 xy + 2 y^{2} + 6 x - 5 y = c$

answer is B.

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Detailed Solution

$\begin{array}{l} \frac{d y}{d x} = \frac{x - 2 y + 3}{2 x - y + 5} \\ C o m p a r i n g w i t h \frac{d y}{d x} = \frac{a_{1} x + b_{1} y + c_{1}}{a_{2} x + b_{2} y + c_{2}} \\ ∴ a_{2} + b_{1} = 0 \\ \Rightarrow (2 x - y + 5) d y = (x - 2 y + 3) d x \\ \Rightarrow 2 x d y - y d y + 5 d y = x d x - 2 y d x + 3 d x \\ \Rightarrow 2 (x d y + y d x) - y d y + 5 d y = x d x + 3 d x \\ \Rightarrow 2 \int D (x y) - \int y d y + \int 5 d y = \int x d x + \int 3 d x \\ \Rightarrow 2 x y - \frac{y^{2}}{2} + 5 y = \frac{x^{2}}{2} + 3 x + c \\ \Rightarrow 4 x y - y^{2} + 10 y = x^{2} + 6 x + c \\ \Rightarrow x^{2} - 4 x y + y^{2} + 6 x - 10 y = c \end{array}$