how to write down the differential equation from a word problem, involving rate of change.

Read the word problem more than once to make sure you fully understand what is being asked and write down the variables involved. Then try to write down all the derivatives in terms of each other, using the chain rule. This will help when it comes to proving an equation being asked. When asked to prove a given equation, you can even work backwards a few steps to get a clue on how you can get to that equation from the ones you already have.