Question

Anton turns his attention to the Australian Financial System. According to his research, Australia has 2000 million dollars of paper money in circulation initially. The Australian government decides to issue new paper money; whenever the old money comes into the banks, it is destroyed and replaced by a equal amount of new money. After 6 weeks, 400 million dollars of NEW paper money are circulating in the country. Let N(t) be the amount of OLD mon (in millions of dollars) in circulation at time t (in weeks). Then N(t) satisfies the differential equation:

dt
dN
​
=kN Part a [2 marks] The order and degree of the above differential equation are respectively. (Enter a number in the boxes) Part b [ 3 marks] How long (in weeks) will it take for 80% of the paper money in circulation to be new paper money?

Answer 1

The order and degree of the differential equation are both 1. The solution to the differential equation is N = Ae^kt, where A is a constant.

Step-by-step explanation:

The order of a differential equation is the highest power of the derivative. In this case, the order of the differential equation is 1.

The degree of a differential equation is the highest power of the highest derivative. In this case, since there is only one derivative term, the degree of the differential equation is also 1.

To find the solution to the differential equation dt dN = kN, we can separate the variables by multiplying both sides by dt and dividing both sides by N. This gives us 1/N dN = k dt. Integrating both sides gives us ln|N| = kt + C, where C is a constant. Solving for N gives us N = e^(kt+C), which simplifies to N = Ae^kt, where A = e^C is another constant.