Differential Equations-Practice 5

Differential Equations

Differential Equations

In this topic, we are looking into transforming some of the real-life physical quantities into mathematical models. The goal is to understand “how fast” a certain quantity change with respect to time.

The typical procedure is to create a mathematical model relating the phenomenon in question. By separating the variables and then perform the integration, we will arrive at the general solution to the problem.

Given a set of initial conditions, we can then create a specific equation relating to the problem, namely the particular solution.

I have put together some of the questions I received in the comment section below. You can try these questions also to further your understanding on this topic.

To check your answer, you can look through the solutions that I have posted either in Youtube videos or Instagram posts.

You can subscribe, like or follow my youtube channel and IG account. I will keep updating my IG daily post, preferably.

Furthermore, you can find some examples and more practices below! =).

Try some of the examples below. You can look at the solution I have written below to study and understand the topic. Cheers ! =) .


EXAMPLE:

1. 9709/03/SP/20 – Specimen Paper 2020 Pure Maths 3 No 10

In a chemical reaction, a compound X is formed from two compounds Y and Z.

The masses in grams of X, Y and Z present at time t seconds after the start of the reaction are x, 10 – x, 20 – x respectively. At any time the rate of formation of X is proportional to the product of the masses of Y and Z present at the time. When t = 0, x = 0 and  dxdt = 2.

(a). Show that x and t satisfy the differential equation

dxdt = 0.01(10  x)(20  x) .

(b). Solve this differential equation and obtain an expression for x in terms of t.


2. 9709/32/M/J/16 – Paper 32 May June 2016 Pure Maths 3 No 6

The variables x and θ satisfy the differential equation

(3 + cos2θ) dxdθ = xsin2θ,

and its given that x = 3 when θ = 14π.

(i). Solve the differential equation and obtain an expression for x in terms of θ.

(ii). State the least value taken by x.


3. 9709/03/SP/17 – Specimen Paper 03 2017 No 8

The variables x and θ satisfy the differential equation

dxdθ = (x + 2) sin22θ,

and its given that x = 0 when θ = 0. Solve the differential equation and calculate the value of x when  θ = 14π, giving your answer correct to 3 significant figures.


4. Compressed air is escaping from a container. The pressure of the air in the container at time t is P, and the constant atmospheric pressure of the air outside the container is A. The rate of decrease of P is proportional to the square root of the pressure difference (PA). Thus the differential equation connecting P and t is

dPdt=k P  A,

where k is a positive constant.

(a). Find the general solution of this differential equation.

(b). Given that P = 5A when t = 0 and that P = 2A when t = 2, show that k = A.

(c). Find the value of t when P = A.

(d). Obtain an expression for P in terms of A and t.


5. The temperature of a quantity of liquid at time t is θ. The liquid is cooling an atmosphere whose temperature is constant and equal to A. The rate of decrease of θ is proportional to the temperature difference (θ  A). Thus θ and t satisfy the differential equation

dθdt=k (θ  A),

where k is a positive constant.

(a). Find the solution of this differential equation, given that  θ = 4A  when t = 0.

(b). Given also that θ = 3A when t = 1, show that k = ln32.

(c). Find θ in terms of A when t = 2, expressing your answer in its simplest form.


6. The variables x and y are related by the differential equation

dydx=6x e3xy2 .

It is given that y = 2 when x = 0. Solve the differential equation and hence find the value of y when x = 0.5, giving your answer correct to 2 decimal places.


7. In the diagram the tangent to a curve at a general point P with coordinates (x, y) meets the x-axis at T. The point N on the x-axis such that PN is perpendicular to the x-axis. The curve is such that, for all values of x in the interval 0<x<12π, the area of triangle PTN is equal to tanx, where x is in radians.

Differential Equations Exercise 4

(i). Using the fact that the gradient of the curve at P is PNTN, show that

dydx=12y2cotx .

(ii). Given that y=2 when x=16π, solve this differential equation to find the equation of the curve, expressing y in terms of x.

8. The variables x and θ satisfy the differential equation

xtanθdxdθ + cosec2θ=0,

for 0<θ<12π  and  x>0. It is given that x = 4 when θ = 16π. Solve the differential equation, obtaining an expression for x in terms of θ.


9. The variables x and y satisfy differential equation

dydx=x ex+y.

It is given that y = 0 when x = 0.

i. Solve the differential equation, obtaining y in terms of x.

ii. Explain why x can only take values that are less than 1.


10. 9709/33/M/J/20 – Paper 33 June 2020 Pure Maths 3 No 4(a), (b)

The equation of a curve is y=x tan1(12x).

(a). Find dydx.

(b). The tangent to the curve at the point where x=2 meets the y-axis at the point with coordinates (0, p).

Find p.


11. 9709/32/M/J/20 – Paper 32 June 2020 Pure Maths 3 No 7

The variables x and y satisfy the differential equation

dydx=y  1(x + 1)(x + 3).

It is given that y=2 when x=0.

Solve the differential equation, obtaining an expression for y in terms of x.


12. 9709/31/M/J/20 – Paper 31 June 2020 Pure Maths 3 No 8(a), (b)

A certain curve is such that its gradient at a point (x,y) is proportional to yxx. The curve passes through the points with coordinates (1,1) and (4,e).

(a). By setting up and solving a differential equation, find the equation of the curve, expressing y in terms of x.

(b). Describe what happens to y as x tends to infinity.


13. 9709/32/F/M/20 – Paper 32 March 2021 Pure Maths 3 No 4(a), (b)

The variables x and y satisfy the differential equation

(1  cosx)dydx = ysinx.

It is given that y=4 when x=π.

(a). Solve the differential equation, obtaining an expression for y in terms of x.

(b). Sketch the graph of y against x for 0<x<2π.


14. 9709/33/M/J/21 – Paper 33 June 2021 Pure Maths 3 No 7(a), (b)

Question 7 Paper 33 June 2021

For the curve shown in the diagram, the normal to the curve at the point P with coordinates (x,y) meets the x-axis at N. The point M is the foot of the perpendicular from P to the x-axis.

The curve is such that for all values of x in the interval 0x<12π, the area of triangle PMN is equal to tanx.

(a)(i). Show that MNy = dydx.

(ii). Hence show that x and y satisfy the differential equation 12y2dydx = tanx.

(b). Given that y=1  when x=0, solve this differential equation to find the equation of the curve, expressing y in terms of x.


15. 9709/32/M/J/20 – Paper 32 June 2020 Pure Maths 3 No 7

A curve is such that the gradient at a general point with coordinates (x,y) is proportional to yx + 1.

The curve passes through the points with coordinates (0,1) and (3,e).

By setting up and solving a differential equation, find the equation of the curve, expressing y in terms of x.


16. 9709/31/M/J/21 – Paper 31 June 2021 Pure Maths 3 No 10

The variables x and t satisfy the differential equation

dxdt = x2(1 + 2x),

and x=1  when t = 0.

Using partial fractions, solve the differential equation, obtaining an expression for t in terms of x.


17. 9709/32/F/M/22 – Paper 32 March 2022 Pure Maths 3 No 9

The variables x and y satisfy the differential equation

(x + 1)(3x + 1)dydx = y,

and it is given that y=1  when x = 1.

Solve the differential equation and find the exact value of y when x = 3, giving your answer in a simplified form.


18. 9709/32/F/M/22 – Paper 32 March 2022 Pure Maths 3 No 9

The variables x and y satisfy the differential equation

(x + 1)(3x + 1)dydx = y,

and it is given that  y = 1  when x = 1.

Solve the differential equation and find the exact value of y when  x = 3, giving your answer in a simplified form.


19. 9709/12/O/N/19 – Paper 12 November 2019 Pure Maths 1 No 3

A curve is such that

dydx = kx,

where  k  is a constant.

The points  P (1,1)  and  Q (4,4)  lie on the curve. Find the equation of the curve.


PRACTICE MORE WITH THESE QUESTIONS BELOW!

1.(i) The number of bacteria in a colony is increasing at a rate that is proportional to the square root of the number of bacteria present. Form a differential equation relating number of bacteria, x, to the time t.

(ii) In another colony the number of bacteria, y, after time t minutes is modelled by the differential equation dydt = 10000y. Find y in terms of t, given that y=900 when t=0. Hence, find the number of bacteria after 10 minutes.

2. A skydiver drops from a helicopter. Before she opens her parachute, her speed v m/s after time t seconds is modelled by the differential equation

dvdt = 10e12t

when t=0, v=0.

(i) Find v in terms of t.

(ii) According to this model, what is the speed of the skydiver in the long term?

She opens her parachute when her speed is 10 m/s. Her speed t seconds after this is w m/s and is modelled by the differential equation dwdt = 12(w  4)(w + 5).

(iii) Express 1(w  4)(w + 5) in partial fractions.

(iv) Using this result show that

w  4w + 5 = 0.4e4.5t.

(v) According to this model, what is the speed of the skydiver in the long term?

3. The number of organisms in a population at time t is denoted by x. Treating x as a continuous variable, the differential equation satisfied by x and t is

dxdt = xetk + et,

where k is a positive constant.

(i) Given that x=10 when t=0, solve the differential equation, obtaining a relation between x, k and t.

(ii) Given also that x=20 when t=1, show that k=1  2e.

(iii) Show that the number of organisms never reaches 48, however large t becomes.

4. In a model of the expansion of a sphere of radius r cm, it is assumed that, at time t seconds after the start, the rate of increase of the surface area of the sphere is proportional to its volume. When t=0, r=5 and drdt= 2.

(i) Show that r satisfies the differential equation

drdt = 0.08r2.

(ii) Solve this differential equation, obtaining an expression for r in terms of t.

(iii) Deduce from your answer to part (ii) the set of values that t can take, according to this model.

5. An underground storage tank is being filled with liquid as shown in the diagram. Initially the tank is empty. At time t hours after filling begins, the volume of liquid is V m3 and the depth of liquid is h m. It is given that V= 43h3.

Differential Equations-Practice 5

The liquid is poured in at a rate of 20 m3 per hour, but owing to leakage, liquid is lost at a rate proportional to h2. When h=1, dhdt= 4.95.

(i) Show that h satisfies the differential equation

dhdt = 5h2  120.

(ii) Verify that

20h2100  h2  20 + 2000(10  h)(10 + h).

(iii) Hence, solve the differential equation in part (i), obtaining an expression for t in terms of h.

6. The variables x and y are related by the differential equation

dydx = 6ye3x2 + e3x.

Given that y=36 when x=0, find an expression for y in terms of x.

7. The variables x and y satisfy the differential equation

(x+1) y dydx = y2 + 5.

It is given that y=2 when x=0. Solve the differential equation obtaining an expression for y2 in terms of x.

8. The coordinates (x, y) of a general point on a curve satisfy the differential equation

x dydx = (2  x)2 y.

The curve passes through the point (1, 1). Find the equation of the curve, obtaining an expression for y in terms of x.

9. The variables x and y satisfy the differential equation

x dydx = (4  y)2,

and y=1 when x=1. Solve the differential equation, obtaining an expression for y2 in terms of x.

10. The variables x and θ satisfy the differential equation

x cos2θ dxdθ = 2tanθ + 1,

for 0θ<12π  and  x>0. It is given that x = 1 when θ = 14π.

(i) Show that

ddθ(tan2θ) = 2tanθcos2θ.

(ii) Solve the differential equation and calculate the value of x when θ = 13π, giving your answer correct to 3 significant figures.


As always, if you have any particular questions to discuss, leave it in the comment section below. Cheers =) .