This is for Optics - the third calc-based physics class I've taken. I'm taking calc4 now and this class seems to use differential equations quite frequently..something i am just now learning. This is an example problem that i don't quite understand: now, the book attempts to explain the solution in the back of the book, but i don't quite get it. it says both a) and b) are waves since they are twice differentiable functions of (z-vt) and (x+vt) respectively. Therefore, for a) psi = a^2(z - bt/a)^2 and the velocity is b/a in the positive z-direction. For b) psi = a^2(x + bt/a + c/a)^2 and the velocity is b/a in the negative x-direction. i don't understand where the bolded equations come from. I understand how they got the velocity and direction from those equations, but no idea how they were derived.