The amount P in a bank account grows continuously at 5% per year. Write the differential equation for P, name the dependent and independent variables, and say why it is a differential equation.
How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write what is given and what is to be found. Given: P grows continuously at 5% of P per year. To find: the DE, the two variables and the reason.
2 · Write the rule. Rate of change of P with time t is the derivative dP/dt.
3 · Translate the statement.
5% of P = (5/100)P = P/20
So dP/dt = P/20
4 · Name the variables. P changes with t, so P is the dependent variable and t (in years) is the independent variable.
5 · Link it to the situation. Banks use this model for interest compounded continuously, so the money grows faster as P grows.
6 · Check and box the answer. dP/dt = P/20. It contains the derivative dP/dt, so it is a differential equation.
Scoring tip: Write 'rate of change of P = dP/dt' as a separate line before the equation. It earns the method mark.
