Systems of differential equations

14 Solve the simultaneous differential equations \(\frac { \mathrm { d } x } { \mathrm {~d} t } + 2 x = 4 y , \quad \frac { \mathrm {~d} y } { \mathrm {~d} t } + 3 x = 5 y\),
given that when \(t = 0 , x = 0\) and \(y = 1\).

An isolated island is populated by rabbits and foxes. At time \(t\) the number of rabbits is \(x\) and the number of foxes is \(y\). It is assumed that: • The number of foxes increases at a rate proportional to the number of rabbits. When there are 200 rabbits the number of foxes is increasing at a rate of 20 foxes per unit period of time. • If there were no foxes present, the number of rabbits would increase by 120% in a unit period of time. • When both foxes and rabbits are present the foxes kill rabbits at a rate that is equal to 110% of the current number of foxes. • At time \(t = 0\), the number of foxes is 20 and the number of rabbits is 80.

1. 1. Construct a mathematical model for the number of rabbits. [9 marks]
   2. Use this model to show that the number of rabbits has doubled after approximately 0.7 units of time. [1 mark]
2. Suggest one way in which the model that you have used for the number of rabbits could be refined. [1 mark]

1. A scientist is studying the effect of introducing a population of type \(A\) bacteria into a population of type \(B\) bacteria.

At time \(t\) days, the number of type \(A\) bacteria, \(x\), and the number of type \(B\) bacteria, \(y\), are modelled by the differential equations $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = x + y \\ & \frac { \mathrm {~d} y } { \mathrm {~d} t } = 3 y - 2 x \end{aligned}$$

1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 0$$
2. Determine a general solution for the number of type \(A\) bacteria at time \(t\) days.
3. Determine a general solution for the number of type \(B\) bacteria at time \(t\) days. The model predicts that, at time \(T\) hours, the number of bacteria in the two populations will be equal. Given that \(x = 100\) and \(y = 275\) when \(t = 0\)
4. determine the value of \(T\), giving your answer to 2 decimal places.
5. Suggest a limitation of the model.

1. Two compounds, \(X\) and \(Y\), are involved in a chemical reaction. The amounts in grams of these compounds, \(t\) minutes after the reaction starts, are \(x\) and \(y\) respectively and are modelled by the differential equations

$$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = - 5 x + 10 y - 30 \\ & \frac { \mathrm {~d} y } { \mathrm {~d} t } = - 2 x + 3 y - 4 \end{aligned}$$

1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 50$$
2. Find, according to the model, a general solution for the amount in grams of compound \(X\) present at time \(t\) minutes.
3. Find, according to the model, a general solution for the amount in grams of compound \(Y\) present at time \(t\) minutes. Given that \(x = 2\) and \(y = 5\) when \(t = 0\)
4. find
   1. the particular solution for \(x\),
   2. the particular solution for \(y\). A scientist thinks that the chemical reaction will have stopped after 8 minutes.
5. Explain whether this is supported by the model.

5 In a predator-prey environment the population, at time \(t\) years, of predators is \(x\) and prey is \(y\). The populations of predators and prey are measured in hundreds. The populations are modelled by the following simultaneous differential equations. \(\frac { \mathrm { d } x } { \mathrm {~d} t } = y \quad \frac { \mathrm {~d} y } { \mathrm {~d} t } = 2 y - 5 x\)

1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } - 5 x\).
2. 1. Find the general solution for \(x\).
   2. Find the equivalent general solution for \(y\). Initially there are 100 predators and 300 prey.
3. Find the particular solutions for \(x\) and \(y\).
4. Determine whether the model predicts that the predators will die out before the prey.

During an industrial process substance \(X\) is converted into substance \(Z\). Some of the substance \(X\) goes through an intermediate phase, and is converted to substance \(Y\), before being converted to substance \(Z\). The situation is modelled by $$\frac{dy}{dt} = 0.3x - 0.2y \quad \text{and} \quad \frac{dz}{dt} = 0.2y + 0.1x$$ where \(x\), \(y\) and \(z\) are the amounts in kg of \(X\), \(Y\) and \(Z\) at time \(t\) hours after the process starts. Initially there is 10 kg of substance \(X\) and nothing of substance \(Y\) and \(Z\). The amount of substance \(X\) decreases exponentially. The initial rate of decrease is 4 kg per hour.

1. Show that \(x = Ae^{-0.4t}\), stating the value of \(A\). [3]
2. Show that \(\frac{dx}{dt} + \frac{dy}{dt} + \frac{dz}{dt} = 0\). Comment on this result in the context of the industrial process. [4]
3. Express \(y\) in terms of \(t\). [5]

1. A scientist is studying the effect of introducing a population of white-clawed crayfish into a population of signal crayfish.
   At time \(t\) years, the number of white-clawed crayfish, \(w\), and the number of signal crayfish, \(s\), are modelled by the differential equations

$$\begin{aligned} & \frac { \mathrm { d } w } { \mathrm {~d} t } = \frac { 5 } { 2 } ( w - s ) \\ & \frac { \mathrm { d } s } { \mathrm {~d} t } = \frac { 2 } { 5 } w - 90 \mathrm { e } ^ { - t } \end{aligned}$$

1. Show that $$2 \frac { \mathrm {~d} ^ { 2 } w } { \mathrm {~d} t ^ { 2 } } - 5 \frac { \mathrm {~d} w } { \mathrm {~d} t } + 2 w = 450 \mathrm { e } ^ { - t }$$
2. Find a general solution for the number of white-clawed crayfish at time \(t\) years.
3. Find a general solution for the number of signal crayfish at time \(t\) years. The model predicts that, at time \(T\) years, the population of white-clawed crayfish will have died out. Given that \(w = 65\) and \(s = 85\) when \(t = 0\)
4. find the value of \(T\), giving your answer to 3 decimal places.
5. Suggest a limitation of the model.

8 A biologist is studying the effect of pesticides on crops. On a certain farm pesticide is regularly applied to a particular crop which grows in soil. Over time, pesticide is transferred between the crop and the soil at a rate which depends on the amount of pesticide in both the crop and the soil. The amount of pesticide in the crop after \(t\) days is \(x\) grams. The amount of pesticide in the soil after \(t\) days is \(y\) grams. Initially, when \(t = 0\), there is no pesticide in either the crop or the soil. At first it is assumed that no pesticide is lost from the system. The biologist further assumes that pesticide is added to the crop at a constant rate of \(k\) grams per day, where \(k > 6\). After collecting some initial data, the biologist suggests that for \(t \geqslant 0\), this situation can be modelled by the following pair of first order linear differential equations. \(\frac { d x } { d t } = - 2 x + 78 y + k\) \(\frac { d y } { d t } = 2 x - 78 y\)

1. 1. Show that \(\frac { d ^ { 2 } x } { d t ^ { 2 } } + 80 \frac { d x } { d t } = 78 \mathrm { k }\).
   2. Determine the particular solution for \(x\) in terms of \(k\) and \(t\). If more than 250 grams of pesticide is found in the crop, then it will fail food safety standards.
   3. The crop is tested 50 days after the pesticide is first added to it. Explain why, according to this model, the crop will fail food safety standards as a result of this test. Further data collection suggests that some pesticide decays in the soil and so is lost from the system. The model is refined in light of this data. The particular solution for \(x\) for this refined model is \(\mathrm { x } = \mathrm { k } \left( 20 - \mathrm { e } ^ { - 41 \mathrm { t } } \left( 20 \cosh ( \sqrt { 1677 } \mathrm { t } ) + \frac { 819 } { \sqrt { 1677 } } \sinh ( \sqrt { 1677 } \mathrm { t } ) \right) \right.\).
2. Given now that \(k < 12\), determine whether the crop will fail food safety standards in the long run according to this refined model. In the refined model, it is still assumed that pesticide is added to the crop at a constant rate.
3. Suggest a reason why it might be more realistic to model the addition of pesticide as not being at a constant rate.