The solution of dydx=x−2y+32x−y+5 is

2xdy−ydy+5dy=xdx−2ydx+3dx→2(xdy+ydx)−ydy+5dy=xdx+3dx→∫2d(xy)−ydy+5dy=∫xdx+3dx→2xy−y22+5y=x22+3x→4xy−y2+10y=x2+6x+c→x2−4xy+y2+6x−10y=c

The solution of dydx=x−2y+32x−y+5 is

2xdy−ydy+5dy=xdx−2ydx+3dx→2(xdy+ydx)−ydy+5dy=xdx+3dx→∫2d(xy)−ydy+5dy=∫xdx+3dx→2xy−y22+5y=x22+3x→4xy−y2+10y=x2+6x+c→x2−4xy+y2+6x−10y=c

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

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

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

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

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

answer is A.

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Detailed Solution

$\begin{array}{l} 2 xdy - ydy + 5 dy = xdx - 2 ydx + 3 dx \\ \to 2 (xdy + ydx) - ydy + 5 dy = xdx + 3 dx \\ \to \int 2 d (xy) - ydy + 5 dy = \int xdx + 3 dx \\ \to 2 xy - \frac{y^{2}}{2} + 5 y = \frac{x^{2}}{2} + 3 x \\ \to 4 xy - y^{2} + 10 y = x^{2} + 6 x + c \\ \to x^{2} - 4 xy + y^{2} + 6 x - 10 y = c \end{array}$