Textbook problem: Consider the equation y' = f(at + by + c), where a, b, and c are constants. Show that the substitution x = at + by + c changes the equation to the separable equation x' = a + bf(x). Use this method to find the general solution of the equation y' = (y + t)^2.
Since nearly everything else given included general functions, I started by implicitly differentiating x = at + by + c, yielding dx = adt + bdy; with respect to t, it follows that x' = a + by', thus x' = a + bf(x), as the problem states. However, I don't understand what this achieves with the goal of somehow applying it to the second part of the question. I also don't understand how I might have discovered the given equation in order to make other, similar non-separable equations separable.
Since nearly everything else given included general functions, I started by implicitly differentiating x = at + by + c, yielding dx = adt + bdy; with respect to t, it follows that x' = a + by', thus x' = a + bf(x), as the problem states. However, I don't understand what this achieves with the goal of somehow applying it to the second part of the question. I also don't understand how I might have discovered the given equation in order to make other, similar non-separable equations separable.