Solve the following differential equation: dx/dt = -2(x-6)^0.5, find t in terms of x given that x=70 when t=0 (6 marks A2)

This question requires a careful and methodical approach. You will start by separating the variables with all x terms on the LHS and all t terms on the RHS. The next step of integration is where most students go wrong. One simple trick to start with is looking for any constants, in this case the -2. These can be brought outside of the integral which reduces the complexity of integration in some instances. So far, we have the (x-6)^-0.5 dx on the LHS and -2 dt on the RHS. By following the rule of increasing the power and dividing by the new power, the LHS is integrated to 2(x-6)^0.5 and the RHS integrated to -2t + c. Most students will forget to include the constant of integration and will skip straight to rearranging to find t. The values provided in the question are a clue to this and by substituting them in to our equation: 2(x-6)^0.5 = -2t + c (x=70 & t=0) we get c = 16. This is then substituted back into the equation and by rearranging, we get our final answer as follows: t = 8 - (x-6)^0.5. Correctly integrating and including the constant of integration are vital to achieve full marks.