In the analysis of homogeneous ordinary differential equations, you can either substitute $y=vx$ or $x=vy$. If $y=vx$, then $$ \frac{\mathrm{d}y}{\mathrm{d}x}= v + x\frac{\mathrm{d}v}{\mathrm{d}x}, $$ and if $x= vy$, then $$ \frac{\mathrm{d}x}{\mathrm{d}y}= v + y\frac{\mathrm{d}v}{\mathrm{d}y}. $$
So can I say that $v+ x\frac{\mathrm{d}v}{\mathrm{d}x} \text{ is equal to } \big(v + y\frac{\mathrm{d}v}{\mathrm{d}y}\big)^{-1}$?
Also, I have found that using $x=vy$ as a substitute gives a much complicated solution than using $y=vx$.
So does this mean that the solution that we get by using each of these substitutions are the same, or are they different solutions? And when should we use $x=vy$ as opposed to $y=vx$?
Well those two expressions are equal, but do notice that v means different things in the different substitution. That is, it’s better to say that y=ux and x=vy. Besides, in differential equations of 1 order, x and y share the same position, generally. But in different situation the complexity may change. It all depends.