The population P of bears in a forest is expected to grow ac) h/a8l5jm) )uh a7 ajia3nlntusd9u1cording to the model tu9auain7d s8h/3j)ha)n jl u1 aml)5

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

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. Theg,;ccef;z 6-to( 3 -dgosgrrk pressure P measured in Pass,tc-(dfr g;6- 3co eor;gkgz cals, and the volume V 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 radium- 226 , a naturally8m( yn:, pznrv8o)cq:9mncip occurring radioactive metal, is directly 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 samz:v8 8norpm,nic(qpy:9) cmnple.
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 money into an account a7b71kr shqy4h,,lkyb -rb0ae t a rate of 1500 euros per year. The account has an annual interest rate of 8 % . The amount of money, A , in the account after 7rq,47bklbkey0 r- , 1bsyhhat 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 3lv8wlxl s)ai8e-y) x equation

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

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 : cb30go+2 1/vsgf)i8ps+rqncyksxkis filled with 2500 $\mathrm{ml}$ 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 differenii5lom l0,b3n tial 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 of a rdfuld/h+ dbt, a:.;2/nse.,jqqe8igqmj i)zectangular 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 shows the slope bn/* b;5e cnsk+4gday 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 .n7wkwju+/u )mul yr* straight line such that its displacement x at time t mku.l n u+)wyru7*/w j$\geq 0$ 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 growth of k4b :)w6o fh6run:vapa penguin colony in a small whkr v n646:)fpo:aubarchipelago 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 population 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 b;0r,q zl . )rk a+p.chkiy:ptof a bee colony in units of 10 is N . At time t weeks, the rate of changrr p lt;c .a0.pbqz,)hy :ik+ke of the population can be modelled by the differential 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 brow9;0 ( chdhpy 5daujfrbvxm h 0-p0x- c-og1sh9n bears in a forest, in units of 100 . At time t months, the rate of change of the pochjph-hpr d sd05ug ybvxao0x9c1 -9hmf(; -0pulation can be modelled by the differential equation

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

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.1/ dlmb cn h31hh33)l,hbna 8h brchx5t3q wk+

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


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 $-8 \leq y \leq 8$ .