If is the amount of heavy fuel and is the amount of light fuel at some point in time and these change by and in time then
In the limit as that gets you
Integrate and substitute the initial condition (to evaluate the constant of integration) and you find that
(Actually, on reflection, you can write this down directly.)
Now substitute that into the expression that you should have written down for
For your solution to the differential equation, you've forgotten to include a constant of integration.