The population P of bears in a forest is expected*cv(dw/3z 4s r(b2iu2ctbz ia to grow according to the modelbs*c(aiu/w i t2( 2cd v3zr4bz

$\frac{\mathrm{d} P}{\mathrm{~d} t}$=$0.2 P\left(1-\frac{P}{2500}\right)$

where t is the time given in years from the beginning of 2020 .
The slope field for this differential equation is as follows:

1. Given that the population at the beginning of 2020 is 100 , sketch the particular solution curve.
2. Find the value at which the population approaches as time increases.
From the slope field we find the population approaches   .

  An experiment is conducted on a gas confined in a container. Th8 g+i)cg13gzws f9e.a1uz4aar jrn1 he pressure P measured in Pascals, and the volume a4 j e.rwgi)z8 1+3zfu1rsg9haganc 1V measured in \mathrm{cm}^{3} satisfy the differential equation

$\frac{\mathrm{d} P}{\mathrm{~d} V}=-\frac{P}{V}$


Initially, the gas has a pressure of 20000 Pascals and it is confined in a container with a volume of 100 $\mathrm{~cm}^{3}$ .
1. Solve the differential equation to show that

P=$\frac{2000000}{V}$

2. Calculate the pressure required to compress the gas to a different container that has a volume of 50 $\mathrm{~cm}^{3}$ .
P=   Pa.

  The decay rate of rad:kldlj qa7;f.,wd+bh ium- 226 , a naturally occurring radioactive metal, is directl; w7.l qakfdb,+ dljh:y proportional to the amount observed at that instant. The half-life of radium-226 is 1600 years. At an initial measurement, there was 100 grams of radium-226 in a sample.
1. Find an expression for the amount of radium-226, R , in the sample in terms of t , where t is the time in years after the initial measurement.
R(t)=ae$^{-0.000433t}$;a=  .
2. Find to the nearest gram the amount of radium- 226 in the sample after 3000 years.
R(3000)≈   grams.(round number)

  Lisa continuously invests mo( fl9 xiv9chw4;wwx3nney into an account at a rate of 1500 euros per year. The account hasx n;fi(vwh9cw x9l 34w an annual interest rate of 8 % . The amount of money, A , in the account after t years satisfies the following differential equation:

$\frac{d A}{d t}=0.08 A+1500$


The initial amount of money Lisa deposited into the account when it was set up was 730 euros.
1. Find an expression for A , in terms of t .
A(t)=  e$^{0.08t}$-  .
2. 1. Find $\int_{0}^{6}(0.08 A+1500) \mathrm{d} t $ ≈   .
2. State what this value represents.

  Solve the differential equatfij:o(0 gnl*:7 rvcjk ion

$\left(1+x^{2}\right) \frac{\mathrm{d} y}{\mathrm{~d} x}=2 x y^{2}$

for y , which satisfies the initial condition y(0)=-$\frac{1}{2}$ .

y=-$\frac{a}{\ln \left(1+x^{2}\right)+b}$;a=  ,b=  .

  A tabletop water purifier is filled with 2j (c.z2)vzc cs500 $\mathrm{ml}$ of water. It takes the purifier 90 seconds to filter 900 $\mathrm{ml}$ of water.
The volume of water, V $\mathrm{ml}$ , remaining unfiltered in the purifier after t seconds, can be modelled by the differential equation

$\frac{\mathrm{d} V}{\mathrm{~d} t}=-q \sqrt{V}$,

where q is a positive constant.
1. Show that V=$\left(50-\frac{t}{9}\right)^{2}$ .

2. Find the time it takes for the purifier to filter all the water.
t=  .

  Consider the differentiawoc0j 1luk+j /pubb ,fy4671zt/ e kuyl equation

$\frac{\mathrm{d} y}{\mathrm{~d} x}$=$\sqrt{x} y, \quad x>0$, y>0 $\text { where }$ y(1)=4 .

1. Use Euler's method with a step length of 0.25 to fill in the following table, rounding each value to two decimal places.

When x is 1.25 , y is   ;When x is 1.50 , y is   ;When x is 1.75 , y is   ;When x is 2 , y is   ;
2. Solve the differential equation.
3. Hence, find the exact value of y(2) to 2 decimal places.
4. Calculate the percentage error in the value of y(2) found by using Euler's method. Give your answer to 2 significant figures.
ϵ≈  %

  A water tank in the shape . ekgrz8i51ed of a rectangular prism has height 4$ \mathrm{~m}$ and a base of 3 $\mathrm{~m} \times 2 \mathrm{~m}$ . Water flows out of a tap at the bottom of the tank at a rate proportional to the square root of the depth of the water at any given time. Let h denote the depth of the water, in metres, and V is the volume of the water remaining in the tank after t minutes.
1. Write down a differential equation for the rate of change of volume of water in terms of time.


2. Show that the volume of water in the tank is given by V=6 $\mathrm{~h} \mathrm{~m}^{3}$ at time t .
V=  h.
3. Hence, or otherwise, show that a differential equation for the rate of change of water height in terms of time is

$\frac{\mathrm{d} h}{\mathrm{~d} t}$=-$\frac{k \sqrt{h}}{6}$

4. Given that the tank is initially full, and the height then drops 3 $\mathrm{~m} $ after 48 minutes, solve the differential equation in part (c).
h=$\left(a-\frac{t}{b}\right)^{2}$;a=  ,b=  .
5. Find the time it takes for the tank to empty.
t=  minutes.

  The diagram below show *ldi)n, ii4vgs the slope field for the differential equation

$\frac{\mathrm{d} y}{\mathrm{~d} x}$=$\cos (x-y$), $\quad-6.5 \leq x \leq 4.5$, $\quad 0 \leq y \leq 5.5$ .


The graphs of the two solutions to the differential equation passing through points P (0,1) and Q(0,3) are drawn over the slope field.

For the two graphs given, the local maximum points lie on the straight line $L_{1} $.
1. Find the equation of $L_{1}$ , giving your answer in the form y=m x+c .
we get y = kx+c$\pi$;k=  ,c=  
For the two graphs given, the local minimum points lie on the straight line $L_{2} $.
2. Find the equation of $L_{2} $, giving your answer in the form y=m x+c .
we get y = kx+c$\pi$;k=  ,c=  

  A particle P moves is a straight line such tj.gy d tqs//t2hat its displacement x at time t/y t/s dg.jt2q $\geq 0$ is given by the differential equation $\dot{x}$=$2 x\left(-t e^{-t^{2}}\right)$ . At time t=0, x=2 .
1. Use Euler's method with step length 0.1 to find an approximation for x when t=0.4 , giving your answer to 4 significant figures.
x(0.4) ≈    (three decimal places)
2. By solving the differential equation, find the percentage error in your approximation for x when t=0.4 .
ϵ≈  %(two decimal places)

  A marine biologist is studying the population gr4nqkv9,2mmx ;1 dy hwxyi. bqg scq2q-r8+k 1zowth of a penguin colony in a small archipelago that is free from predators. At the start of her study, there was estimated to be 200 penguins in the colony. It is assumed the populati- dmqyrb12 z+iv9 8mx2kqyx kgs1q4qwc,hn;.on growth of penguins can be modelled by the differential equation

$\frac{\mathrm{d} P}{\mathrm{~d} t}$=1.1 P

where P is the penguin population at time t years.
1. Find the population of penguins after 3 years.
After 3 years there are approximately    penguins
​

When the population of penguins reaches 5000 , it is noticed that a group of 100 leopard seals have settled in the area. The subsequent population growth of penguins and leopard seals, where L is the population of leopard seals at time t , can be modelled by the coupled differential equations

$\frac{\mathrm{d} P}{\mathrm{~d} t}$=P(2.2-0.011 L) $\quad \frac{\mathrm{d} L}{\mathrm{~d} t}$=$L(0.0002 P-0.7)$

2. Using Euler's method with a step size of 0.25 , estimate
1. the population of penguins 1 year after the leopard seals were noticed;
After 1 year there are approximately    penguins
2. the population of leopard seals 1 year after they were noticed.
After 1 year there are approximately    leopard seals
​

The graph of the population sizes, according to this model, for the first 4 years after the leopard seals were noticed is shown below.

3. Describe the changes in the populations of penguins and leopard seals for these 4 years,
1. at point A ;
2. at point B .
4. Find the non-zero equilibrium point for the populations of penguins and leopard seals.
therefore the non-zero equilibrium point is
P(  ,  )
​

  Suppose the population size of a bee colony in units i e3rn 86ewlv.1gz /rtu1msu d+c9m 8uif4i h)of 10 is N . At time t weeks, the rate of change of the population can be modelled by the diffe muci v 1si+4/lr1zn.8u3)f8te imwhuderg96rential equation $\frac{\mathrm{d} N}{\mathrm{~d} t}$=0.4 N-0.8 t
1. Given that N=a+b t , for a, b $\in \mathbb{R}$ , is a solution to the differential equation for a particular initial population, find the values of a and b .
a=  .
The slope field for the differential equation is shown below

2. Sketch on the slope diagram:
1. the line N=a+b t
2. the trajectory of the population if at t=0, N=3 .
3. Find the least value for N at t=0 that will ensure the population does not become extinct.
N=  .
A beekeeper measuring the population N determines it will reach a maximum after two and a half weeks and then will begin to decline.
4. Write down an approximation for N at that time.

The beekeeper decides to introduce more bees at t=2.5 .
5. If the model remains valid, find the least number of bees N that needs to be added in order for the population to continue to increase in size as time increases.
Therefore, the beekeeper needs to increase N by   .
Suppose that N=80 after 4 weeks.
6. Estimate N after 5 weeks by using Euler's method with a step size of 0.2 .
N≈  .

  N is the population size of brown bears in anzou vl;j6xsi.x v(; / forest, in units of 100 . At time t months, the rate of change of the population can be modelled by the differential equation /(vv; li jzou.x nx6s;

$\frac{\mathrm{d} N}{\mathrm{~d} t}=0.3 N-0.72 t$

1. Given that N=a+b t , for a, b $\in \mathbb{R}$ , is a solution to the differential equation 4 for a particular initial population, find the values of a and b .
a=  ,b=  .
The slope field for the differential equation is shown below


2. Sketch on the slope diagram:
1. the line N=a+b t
2. the trajectory of the population if at t=0, N=6 .

3. Find the least value for N at t=0 that will ensure the population does not become extinct.
Therefore the minimum initial value for which population does not become extinct is N=  .
​

A group of conservationists concerned with the brown bears extinction determines that the brown bear population will reach a maximum after six months and then begin to decline.

4. Write down an approximation for N at that time.

The conservationists decide to introduce more bears at t=6 months.we conclude the population at t=6 is   .
​

5. If the model remains valid, find the least number of bears needed to be added for the population to continue to increase in the future.

Suppose that N=28 after 8 months.
6. Estimate N after 9 months by using Euler's method with a step size of 0.2 .
after 9 months
N≈  .(one decimal place)
​

  Consider the following system of coupled differential equations. .l13svo;bc8 kpdw.uy

$\begin{array}{l}
\frac{\mathrm{d} x}{\mathrm{~d} t}=2 x+3 y \\
\frac{\mathrm{d} y}{\mathrm{~d} t}=2 x+y
\end{array}$


The system can be written in the form



where A is a $2 \times$ 2 matrix.
1. 1. Write down matrix A .

A=$ \begin{Bmatrix} a & b \\ c & d \end{Bmatrix} $
a=  ,b=  ,c=  ,d=  .
2. Find the eigenvalues and corresponding eigenvectors of matrix A .
2. Hence write down the general solution of the system.

$X$=A e$^{-t}$$ \begin{Bmatrix} a \\ -1 \end{Bmatrix} $+B $e^{4 t}$$\begin{Bmatrix} b \\ 2 \end{Bmatrix} $
a=  ,b=  .
3. Determine whether the equilibrium point E(0,0) is stable or unstable. Justify your answer.
4. Find the value of $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at point:
1. $ \mathrm{P}(5,0) $ is   .
2. $ \mathrm{Q}(-5,0)$ is   .
5. Sketch a phase portrait for the general solution to the system of coupled diff erential equations for $-8 \leq x \leq 8$ and $-8 \leq y \leq 8$ .