Consider the differentin9v ortqh on r:y0o +8ba9h-*zal equation $\frac{\mathrm{d} y}{\mathrm{~d} x}-e^{x}=1$
Given that y(0)=1 , use Euler's method with step length h=0.25 to find an approximation for y(1) . Give your answer correct to two decimal places.

≈   

  Solve the differential equation dtiid7/kfm .3/d)rew

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

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

  Consider the differe yte/qqsmm 5vm9mzj63- ku2kbe9 0t/dntial equation $ \frac{\mathrm{d} y}{\mathrm{~d} x}+\left(\frac{6 x}{3 x^{2}-2}\right) y=4 x$ , given that y=4 when x=0 .
1. Show that $3 x^{2}-2$ is an integrating factor for this differential equation.  (代数式) 
2. Hence solve this differential equation. Give the answer in the form y=f(x) .  (代数式) 

  There is a rumour spreading about thedar 0i1 z5 :vl54aw6o p3a zp-f*trcse questions that will appear in an upcoming chemistry exam in a class with a large number of students. Let x be the proportion of students who have heard the rumor and let t be the time in3p6:c r0od4 -1afzir*pvat z55awesl hours, after 10.00 a.m.

The situation can be modelled by the differential equation $ \frac{\mathrm{d} x}{\mathrm{~d} t}=k x(1-x)$ where k is a constant.
1. Use partial fractions to solve this differential equation and hence show that $\frac{x}{1-x}=A e^{k t} $, where A is a constant  (代数式) 
2. At 10.00 $\mathrm{a}$ .$ \mathrm{m}$ . one tenth of the students know about the rumour. Find the value of A   
3. At 12.00 p.m., the proportion of students who knew about the rumor is 0.55 . Find the value of k ≈   
4. Hence, find the proportion of students who knew about the rumour at 1.00 p.m.≈   

Solve the differential equatini1 3u4 a9pifxg( :4ldqxgl v*9jd9dj higz;-on

$\sqrt{1-x^{2}} \frac{\mathrm{d} y}{\mathrm{~d} x}=\sqrt{1-y^{2}}$

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

1. Show that $y=\frac{1}{2 x^{2}} \int f(x) \mathrm{d} x$ is a solution of the differential equation

$2 x^{2} \frac{\mathrm{d} y}{\mathrm{~d} x}+4 x y=f(x)$ .

2. Hence solve $2 x^{2} \frac{\mathrm{d} y}{\mathrm{~d} x}+4 x y=\frac{1}{x}$, x>0 , given that y=2 when x=1 .

  Consider the differentii irm8)-k 9ujcal equation $\frac{\mathrm{d} y}{\mathrm{~d} x}-\frac{y}{x}=\frac{1}{2}$ , where x>0 .
1. Given that y(1)=2 , use Euler's method with step length h=0.5 to find an approximation for y(3) . Give your answer correct to two significant figures.≈   
2. Solve the equation $ \frac{\mathrm{d} y}{\mathrm{~d} x}-\frac{y}{x}=\frac{1}{2}$ for y(1)=2 .  (代数式) 
3. Find the percentage error when y(3) is approximated by the final rounded value found in part (a). Give your answer correct to two significant figures.≈    %

Consider the differenti o(wn7y(ntfuv /n3 oo8al equation

$x^{2} \frac{\mathrm{d} y}{\mathrm{~d} x}+6 x^{2}=y^{2}$

for x>0 and y>3 x . It is given that y=4 when x=1 .
1. Use Euler's method, with a step length of 0.08 , to find an approximate value for y when x=1.4 .
2. Use the substitution y=v x to show that $x \frac{\mathrm{d} v}{\mathrm{~d} x}=v^{2}-v-6 $.
3. By solving the differential equation, show that $y=\frac{18 x+2 x^{6}}{6-x^{5}}$ .
4. 1. Find the actual value of y when x=1.4 .
2. Using the graph of $ y=\frac{18 x+2 x^{6}}{6-x^{5}} $, suggest a reason why the approximation given by Euler's method in part (a) is not a good estimate to the actual value of y at x=1.4

Consider the differentialgf 5vxkg*acz 7 uicx()6cvt 30 equation $x \frac{\mathrm{d} y}{\mathrm{~d} x}+y=x^{p+1} $ where $x \in \mathbb{R}$, $x \neq 0 $ and p is a positive integer, p>0 .
1. Solve the differential equation given that y=1 when x=1 . Give your answer in the form y=f(x) .
2. 1. Show that the x -coordinate(s) of the points on the curve y=f(x) where $\frac{\mathrm{d} y}{\mathrm{~d} x}=0 $ satisfy the equation $x^{p+2}=1$ .
2. Deduce the set of values for p such that there are two points on the curve y=f(x) where $ \frac{\mathrm{d} y}{\mathrm{~d} x}=0 $. Give a reason for your answer.

  Consider the differentialq7bwh58u uta.o/ wgo- equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{2 x^{3}}{1+x^{4}} y=4 x^{3} $ where y=1 when x=0 .
1. Show that $\sqrt{1+x^{4}} $ is an integrating factor for this differential equation.  (代数式) 
2. Solve the differential equation giving your answer in the form y=f(x) .  (代数式) 

1. Consider the differential equatxro+6p ,idgv.ps 5gaaa44 v * l6ns)uyion

$\frac{\mathrm{d} y}{\mathrm{~d} x}=f\left(\frac{y}{x}\right), \quad x\lt 0$ .

Use the substitution$ v=\frac{y}{x}$ to show that the general solution of this differential equation is

$\int \frac{\mathrm{d} v}{f(v)-v}=\ln x+C$

2. Hence, or otherwise, solve the differential equation

$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{4 x^{2}+5 x y+y^{2}}{x^{2}}, \quad x\lt 0$,

given that y=2 when x=1 . Give your answer in the form y=g(x) .

Consider the differen amgh0) sj;wvgze(wqi 9)3/rt tial equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{x}{3 y^{2}}+x y $, where y=1 when x=0 .
1. Show that $ z=y^{3}$ transforms the differential equation into $ \frac{\mathrm{d} z}{\mathrm{~d} x}-3 x z=x $.
2. By solving this differential equation in z , obtain an expression for y in terms of x .

The curves y=f(x) and y=g(+ d:(sr2;makrwo dm7hx) both pass through the point (1,0) and are defined by the differentiah:7k(d2 ; od mmr+asrwl equations $ \frac{\mathrm{d} y}{\mathrm{~d} x}=2 x-y^{2}$ and $\frac{\mathrm{d} y}{\mathrm{~d} x}=3 y-\frac{x}{2} $ respectively.
1. Show that the tangent to the curve y=f(x) at the point (1,0) is normal to the curve y=g(x) at the point (1,0) .
2. Find g(x) .
3. Use Euler's method with steps of 0.2 to estimate f(2) to 5 decimal places.

Consider the differentia:48ppmi28i a3uqxop t jm+z0pl equation $ \frac{\mathrm{d} y}{\mathrm{~d} x}-(\tan x) y=1$ , where $x \neq \frac{(2 n+1) \pi}{2} $, for any integer n .
1. Given that y(0)=1 , use Euler's method with step length h=0.2 to find an approximation for y(1) . Give your answer correct to two decimal places.
2. Solve the equation $ \frac{\mathrm{d} y}{\mathrm{~d} x}-(\tan x) y=1 $. Give your answer in the form y=f(x) .
3. Find the percentage error when y(1) is approximated by the final rounded value found in part (a). Give your answer correct to two significant figures.
4. Show that the x -coordinate(s) of the points on the curve y=f(x) where $ \frac{\mathrm{d} y}{\mathrm{~d} x}=0 $ are of the form $x=\frac{1}{2}(4 \pi n-\pi)$ , where $ n \in \mathbb{Z}$ .

The acceleration, $a \mathrm{~ms}^{-2} $ of a particle moving in a vertical trajectory at time t seconds, $ t \geq 0$ , is given by $a(t)=-(3+v) $ where v is the particle's velocity in $\mathrm{ms}^{-1} $. At t=0 , the particle is at a fixed origin O and has an initial velocity of $v_{0} \mathrm{~ms}^{-1}$ .
1. By solving an appropriate differential equation, show that the particle's velocity is given by v$(t)=\left(v_{0}+3\right) e^{-t}-3 $.

The particle initially moves upwards until it reaches its maximum height from O , and then returns to O .
Let s metres represent the particle's displacement from O , and $s_{\max }$ the maximum displacement from O .
2. 1. Show that the time T taken for the particle to reach$ s_{\max } $ satisfies the equation $e^{-T}=\frac{3}{v_{0}+3} $.
2. Hence, solve for T in terms of $ v_{0}$ .
3. By solving an appropriate differential equation and using the results from part (b) (i) and (ii), find an expression for $s_{\text {max }} $ in terms of $ v_{0}$ .

Let v(T-k) represent the particle's velocity k seconds before it reaches $ s_{\max }$ , where

$v(T-k)=\left(v_{0}+3\right) e^{-(T-k)}-3$

3. By using the result from part (b) (i), show that v(T-k)=3 e^{k}-3 .

Similarly, let v(T+k) represent the particle's velocity k seconds after it reaches $s_{\text {max }}$
4. Deduce a similar expression for v(T+k) in terms of k .
5. Hence, show that $ v(T-k)+v(T+k) \geq 0 $.

A video streaming service company 0zoaq,*hfn 2c.v/b*g 4 szaqiare monitoring their market share in a region inhsaqc24a/*gz* f0. nq,bzv i o which they have recently commenced operations.
The number of households, N , they predict will subscribe to the streaming service can be modelled by the logistic differential equation

$\frac{\mathrm{d} N}{\mathrm{~d} t}=\frac{3 k N(L-N)}{2 L}$

where t is time measured in years and k, L are positive constants.
The constant L represents the total number of households in the region who could possibly subscribe to the streaming service.
1. Show that $\frac{\mathrm{d}^{2} N}{\mathrm{~d} t^{2}}=\left(\frac{3 k}{2 L}\right)^{2}(N)(L-N)(L-2 N)$ .
2. Hence show that the number of households subscribing to the streaming service is predicted to increase at its maximum rate when $N=\frac{L}{2}$ .
3. Hence determine the maximum value of $\frac{\mathrm{d} N}{\mathrm{~d} t}$ in terms of k and L .

Let N_{0} be the number of households who have subscribed to the streaming service at the time the company start monitoring their market share.
4. By solving the logistic differential equation, show that its solution can be expressed in the form

k$ t=\left(\frac{2}{3}\right) \ln \left(\frac{N\left(L-N_{0}\right)}{N_{0}(L-N)}\right)$

After 12 years, the number of subscribed households is predicted to be 4$ N_{0} $. It is known that L=5$ N_{0} $.
5. Find the value of k for this model.

A tank has been prepared in a x)l ,rh:mu3 75xvczdorder to mix a color for a fabric dyeing process. The tank initially contains water. A color )h a5z3lmd7x,c :vxurconcentrate is a premix of color powder and a small amount of water The color concentrate begins to flow into the tank. The color solution is kept uniform by stirring and leaves the tank through an outlet at its base. Let x grams represent the amount of the color powder in the tank and let t minutes represent the time since the color concentrate began flowing into the tank.
The rate of change of the amount of color powder in the tank, $\frac{\mathrm{d} x}{\mathrm{~d} t}$ , is described by the differential equation

$\frac{\mathrm{d} x}{\mathrm{~d} t}=4 e^{-\frac{t}{5}}-\frac{x}{t+3}$

1. Show that t+3 is an integrating factor for this differential equation.
2. Hence, by solving this differential equation, show that $ x(t)=\frac{160-20 e^{-\frac{t}{5}}(t+8)}{t+3}$ .
3. Sketch the graph of x versus t for $0 \leq t \leq 50 $ and hence find the maximum amount of color powder in the tank and the value of t at which this occurs.
4. Find the value of t at which the amount of color powder in the tank is decreasing most rapidly.

The rate of change of the amount of color powder leaving the tank is equal to
5. Find the amount of color powder that left the tank during the first 50 minutes.

The population P of fish in a lake after t weeks can be njk f39(baa;+id5 q2q0q x ovzmodelled by the differdjx+ z 2vbi;a5qq 0f 3nqo9ak(ential equation.

$\frac{\mathrm{d} P}{\mathrm{~d} t}=k \sqrt{P}, \quad k, t>0$

1. Show that the population of fish is given by

$P(t)=\left(\frac{k t}{2}+\sqrt{P_{0}}\right)^{2}, \quad t>0$

where $ P_{0}$ is the initial fish population.
It is known that the initial fish population was 3000 , and that 24 weeks later the population had doubled in size.
2. Find the value of k to three significant figures.
3. Estimate the number of fish after 30 weeks to the nearest integer.

After a careful adjustment it is found that the model that best describes the fish population is given by

$\frac{\mathrm{d} P_{2}}{\mathrm{~d} t}=(1.89+3 \cos (0.2 \pi t)) \sqrt{P_{2}}$

where t is the time measured in weeks, $t \geq 0 $.
4. Verify that $ P_{2}=\left(\frac{1.89 t}{2}+\frac{30 \sin (0.2 \pi t)}{4 \pi}+\sqrt{3000}\right)^{2} $ is the solution of this new differential equation.
5. Sketch the graph of P_{2}(t) and the graph of P(t) found in parts (a) and (b) on the same axes, for $ 0 \leq t \leq 50 $.
6. Use $ P_{2}(t) $ to estimate the number of whole weeks it takes for the population to reach 5000 fish.