Question

The solution of the differential equation $(x-y)(2dy-dx)=3dx-5dy$ is:

• A. $2x-y=\log (x-y+z)+c$
• B. $2x+y=\log (x-y+z)+c$
• C. $2y-x=\log (x-y+z)+c$
• D. None of these

Answer 1

The correct option is C $2y-x=\log (x-y+z)+c$

The given equation can be written as
$(2x-2y+5)dy=(x-y+3)dx$
$\Rightarrow \frac{dy}{dx}=\frac{x-y+3}{2(x-y)+5}$
Put $x-y=V\Rightarrow 1-\frac{dy}{dx}=\frac{dV}{dx}$
Therefore, the given equation becomes
$1-\frac{dV}{dx}=\frac{V+3}{2V+5}$ $\Rightarrow \frac{dV}{dx}=\frac{V+2}{2V+5}$
$\Rightarrow dx=\frac{2V+5}{V+2}dV$ $\Rightarrow dx=(2+\frac{1}{V+2})dV$
On integrating, we get $x=2V+\log (V+2)+c$
$\Rightarrow x=2(x-y)+\log (x-y+2)+c$
Therefore, $2y-x=\log (x-y+2)+c,$ is the required solution.