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.
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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:
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 (P – A). 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.
(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 tan−1(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 yx√x. 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)
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 0≤x<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 y√x + 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 = k√x,
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 = 10000√y. 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 = 10e−12t
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 = 0.4e−4.5t.
(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 = xe−tk + e−t,
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.
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 0≤θ<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 =) .