differential equation (dy/dx) = sin(y - x) is given show that if a new variable is defined by y = z^(1-n) then the equation (dz/dx) + p(x)z = q(x)z^n becomes a linear differential equation in y(x)
Then just do it.
We have $\displaystyle \frac{dz}{dx}\cdot \frac{1}{{{z}^{n}}}+\frac{p(x)}{{{z}^{n-1}}}=q(x)$ which is a Bernoulli ODE so put $\displaystyle y=\frac1{z^{n-1}}=z^{1-n}$ (and note that this is the suggested substitution) and you'll turn that ODE into a linear one.