The population P of bears in a forest w x qiio ;t)1wuq3i gp51u/f1* cdt/dkis expected to grow according to the modecq1xi ig3w1qkui/u/d5f1d;p )t t wo*l

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

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 conductedg 7nrj: 3qnum1)udus/ on a gas confined in a container. The pressure P measured in Pascals, and d/ qsj1n: nu)7muurg 3the 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 ,:(fevqon l+1x a naturally occurring radioactive metal, is directly proportional to the amount observed at that instant. Thn el1f:+(x qove 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 money into an account at a rf/n: 7 uc; ao:8b izimpeky1,(nxpcz )ate of 1500 euros per year. The account has apm;i c c xnoizpze:nb7(y8k),/a1f :u n annual interest rate of 8 % . The amount of money, A , in the account after t years satisfies the following differential equation:

$\frac{dA}{dt}=0.08A+1500$


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 ${\int }_0^6(0.08A+1500)\mathrm{d}t$ ≈   .
2. State what this value represents.

  Solve the differential,x0khlw1btxoc(51i x equation

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

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

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

  A tabletop water purifier is fillif ; oanv+eojxm1l8 .i0mzgkg0/j9v -ed 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=${(50-\frac{t}{9})}^2$ .

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

  Consider the differential equatqet nlx6fawlpy*8*3 5 l(.nf zion

$\frac{\mathrm{d}y}{\mathrm{d}x}$=$\sqrt{x}y,x>0$, 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 rectangular pri vvx7 ,)n32hu gsfzm9tsm 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=${(a-\frac{t}{b})}^2$;a=  ,b=  .
5. Find the time it takes for the tank to empty.
t=  minutes.

  The diagram below shows the slope field for the differenti lky4: dkat68hlfl. 6bal equation

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


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 s wcg yjq8qn8q*i j..2:eaal ;mtraight line such that its displacement x at time qmc*q2e.nj.wqyjg8;lai : 8a t $\ge 0$ is given by the differential equation $\dot{x}$=$2x(-t{e}^{-t^2})$ . 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 f8d 6it3h9c r*:bdy6z j4phkwthe population growth of a penguin colony in a small archipelago that is free from predators. At the start of her study, there was estimated to b 8dpjiz6:9 h6r4*b ywhkc ft3de 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) $\frac{\mathrm{d}L}{\mathrm{d}t}$=$L(0.0002P-0.7)$

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 of 10 iwau yyngctt3m hb bw2kl ;z, .10*)or1s N . At time t weeks, the rate of change of the population can be modelnzkwgw )*c0 t amr.ly h; u31y21,botbled 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 brown bears in a forest, in units of 1gf 37ux r2k ta3)c.rvf00 . At time t months, th fua7r.tkv) gc3xfr 23e rate of change of the population can be modelled by the differential equation

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

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 c+qr 2oa1 c)ocioupled differential equations.

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


The system can be written in the form



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

A=$\left\{\begin{array}{cc}a& b\\ c& d\end{array}\right\}$
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}$$\left\{\begin{array}{c}a\\ -1\end{array}\right\}$+B $e^{4t}$$\left\{\begin{array}{c}b\\ 2\end{array}\right\}$
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 $-8\le x\le 8$ and $-8\le y\le 8$ .