# CIE A level -Pure Mathematics 3 :Topic : 3.8 Differential equations :solution for a ﬁrst order differential equation: Exam Style Questions Paper 3

### Question

The coordinates (x, y )of a general point on a curve satisfy the differential equation$$x\frac{dy}{dx}=(2-x^{2})y$$

The curve passes through the point (1, 1). Find the equation of the curve, obtaining an expression for
y in terms of x.                                                                                                                                                                                                                        [7]

Separate variables correctly and integrate at least one side Obtain term ln y

Obtain terms 2 ln  $$x-\frac{1}{2}x^{2}$$

Use x = 1, y = 1 to evaluate a constant, or as limits

Obtain correct solution in any form, e.g.$$In y=2 In x-\frac{1}{2}x^{2}+\frac{1}{2}$$

Rearrange as y=$$x^{2} exp (\frac{1}{2}-\frac{1}{2}x^{2})$$  or equivalent

### Question

The number of insects in a population t weeks after the start of observations is denoted by N. The population is decreasing at a rate proportional to $$Ne^{-0.02t}$$The variables N and t are treated as continuous, and it is given that when t = 0, N = 1000 and $$\frac{dN}{dt}$$=-10

(i) Show that N and t satisfy the differential equation $$\frac{dN}{dt}=-0.01e^{-0.02t}N$$

(ii) Solve the differential equation and find the value of t when N = 800.

(iii) State what happens to the value of N as t becomes large.

(i) State$$\frac{dN}{dt}=ke^{-0.02t}$$ N − and show k = – 0.01

(ii) Separate variables correctly and integrate at least one side

Obtain term ln N

Obtain term$$0.5e^{-0.02t}$$

Use N = 1000, t = 0 to evaluate a constant, or as limits, in a solution with terms a N ln and $$be^{-0.02t}$$ , where ab
≠ 0

Obtain correct solution in any form e.g. In N -1000= $$0.5(e^{-0.02t}-1)$$ Substitute N = 800 and obtain t = 29.6

(iii) State that N approaches $$\frac{1000}{\sqrt{e}}$$

### Question

(i) Differentiate $$\frac{1}{sin^{2}\Theta }$$with respect to$$\Theta$$

(ii) The variables x and 1 satisfy the differential equation$$xtan\Theta \frac{dx}{d\Theta }+cosec^{2}$$=0,

for 0 < 1 <$$\frac{1}{2}$$and x > 0. It is given that x = 4 when $$\Theta \frac{1}{6}\pi$$.Solve the differential equation, obtaining an expression for x in terms of $$\Theta$$

(i) Use chain rule Obtain correct answer in any form

(ii) Separate variables correctly and integrate at least one side Obtain term $$\frac{1}{2}x^{2}$$

Obtain term of the form $$\frac{k}{sin^{2}\Theta }$$

Use x = 4,$$\Theta =\frac{1}{6\Pi }$$ to evaluate a constant, or as limits, in a solution with terms and  $$ax^{2} and \frac{b}{sin^{2}\Theta }$$ where ab
≠ 0 Obtain solution $$\sqrt{(cosec^{2}+12)}$$

### Question

The variables x and y are related by the differential equation

$$\frac{dy}{dx}=\frac{6ye^{3x}}{2+e^{3x}}$$

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

Separate variables correctly and recognisable attempt at integration of at least one side
Obtain lny, or equivalent

Obtain

Use y(0) = 36 to find constant in

Obtain equation correctly without logarithms from

Obtain

### Question

A tank containing water is in the form of a cone with vertex C. The axis is vertical and the semivertical angle is 60°, as shown in the diagram. At time t = 0, the tank is full and the depth of water is H. At this instant, a tap at C is opened and water begins to flow out. The volume of water in the tank decreases at a rate proportional to √h, where h is the depth of water at time t. The tank becomes empty when t = 60.

(i) Show that h and t satisfy a differential equation of the form

$$\frac{dh}{dt}=-Ah^{-\frac{3}{2}},$$

where A is a positive constant.[4]

(ii) Solve the differential equation given in part (i) and obtain an expression for t in terms of h and H.[6]

(iii) Find the time at which the depth reaches $$\frac{1}{2}H$$.[1]

[The volume V of a cone of vertical height h and base radius r is given by $$V=\frac{1}{3}\pi r^{2}h.$$ ]

Ans:

### Question

Liquid is flowing into a small tank which has a leak. Initially the tank is empty and, t minutes later, the volume of liquid in the tank is V cm3. The liquid is flowing into the tank at a constant rate of 80 cm3 per minute. Because of the leak, liquid is being lost from the tank at a rate which, at any instant, is equal to kV cm3 per minute where k is a positive constant.

(i) Write down a differential equation describing this situation and solve it to show that [7]

$$V=\frac{1}{k}\left ( 80-80e^{-kt} \right )$$

(ii) It is observed that V = 500 when t = 15, so that k satisfies the equation

$$k=\frac{4-4e^{-kt} }{25}$$

Use an iterative formula, based on this equation, to find the value of k correct to 2 significant figures. Use an initial value of k = 0.1 and show the result of each iteration to 4 significant figures. [3]

(iii) Determine how much liquid there is in the tank 20 minutes after the liquid started flowing, and state what happens to the volume of liquid in the tank after a long time.[2]

Ans:

(i) State $$\frac{dV}{dt}=80-kV$$
Correctly separate variables and attempt integration of one side
Obtain a ln(80 − kV) = t or equivalent
Obtain $$-\frac{1}{k}ln\left ( 80-kV \right )=t$$ or equivalent
Use t = 0 and V = 0 to find constant of integration or as limits
Obtain $$-\frac{1}{k}ln\left ( 80-kV \right )=t-\frac{1}{k}$$ or equivalent
Obtain given answer $$V=\frac{1}{k}\left ( 80-80e^{-kt} \right )$$ correctly

(ii) Use iterative formula correctly at least once
Obtain final answer 0.14
Show sufficient iterations to 4 s.f. to justify answer to 2 s.f. or show a sign change in the interval (0.135, 0.145)

(iii) State a value between 530 and 540 cm3 inclusive
State or imply that volume approaches 569 cm3 (allowing any value between 567 and 571 inclusive)

### Question

The variables x and y satisfy the differential equation $$\frac{dy}{dx}=4cos^{2}ytanx.for 0\leqslant x< \frac{1}{2}\pi , and x=0 when y=\frac{1}{4}\pi$$,Solve this differential equation and find the value of x when $$\frac{1}{3}\pi$$

Separate variables correctly and attempt integration of one side  Obtain term tan y , or equivalent
Obtain term of the form k x ln cos , or equivalent
Obtain term −4ln cos x , or equivalent  Use x = 0 and  y =$$\frac{1}{4} π$$ in solution containing a y tan and b x ln cos to evaluate a constant, or as limits Obtain correct solution in any form, e.g. tan 4ln secx+1  Substitute y =$$\frac{1}{3} π$$  in solution containing terms a y tan and b x ln cos , and use correct
method to find x Obtain answer x = 0.587

### Question

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

$$x\frac{dy}{dx}=(1-2x^{2})y,$$

for x > 0. It is given that y = 1 when x = 1.
Solve the differential equation, obtaining an expression for y in terms of x.                                             [6]

Separate variables correctly and attempt integration of at least
one side
Obtain term ln y
Obtain terms 2 ln x − x2
Use x = 1, y = 1 to evaluate a constant, or as limits, in a solution
containing at least 2 terms of the form a ln  y , b  ln x and cx
Obtain correct solution in any form
Rearrange and obtain $$y=xe^{1-x^{2}}$$

### Question

A certain curve is such that its gradient at a point (x, y) is proportional to $$\frac{y}{x\sqrt{x}}$$. 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.                                                                                                                                                                          [8]

(b) Describe what happens to y as x tends to infinity.                                                                                                     [1]

Ans

(a) State $$\frac{dy}{dx}=k\frac{y}{x\sqrt{x}}$$, or equivalent

Separate variables correctly and attempt integration of at least one side M1
Obtain term ln y, or equivalent
Obtain term $$-2k\frac{1}{\sqrt{x}}$$, or equivalent

Use given coordinates to find k or a constant of integration c in a solution containing terms of the form a ln y and $$\frac{b}{\sqrt{x}}$$,  where ab≠ 0
Obtain k = 1 and c = 2

Obtain final answer y = exp $$\left ( -\frac{2}{\sqrt{x}}+2 \right )$$, or equivalent

(b) State that y approaches e2
(FT their c in part (a) of the correct form)

### Question

The variables x and y satisfy the differential equation
$$\frac{dy}{dx}=\frac{1+4y^2}{e^x}$$.
It is given that y=0 when x = 1.
(a) Solve the differential equation, obtaining an expression for y in terms of x.
(b) State what happens to the value of y as x tends to infinity.

Ans:

(a) Separate variables correctly and attempt integration of at least one side
Obtain term of the form $$atan^{-1}(2y)$$
Obtain term $$\frac{1}{2}tan^{-1}(2y)$$
Obtain term $$-e^{-x}$$
Use x = 1, y = 0 to evaluate a constant or as limits in a solution containing
terms of the form $$atan^{-1}(by)$$ and $$ce^{\pm x}$$
Obtain correct answer in any form
Obtain final answer $$y=\frac{1}{2}tan(2e^{-1}-2e^{-x})$$, or equivalent
(b) State that y approaches $$\frac{1}{2}tan(2e^{-1})$$, or equivalent

### Question

(a)Given that y = ln ( ln x), show that

$$\frac{dy}{dx} = \frac{1}{x ln x}.$$

The variables x and t satisfy the differential equation

x ln x + t $$\frac{dx}{dt} = 0$$

It is given that x = e when t = 2.

(b)Solve the differential equation obtaining an expression for x in terms of t, simplifying your answer.

(c)Hence state what happens to the value of x as t tends to infinity.

Ans:

(a)Show sufficient working to justify the given answer

(b)Correct separation of variables

Obtain term ln  ( ln x)

Obtain term −ln t

Evaluate a constant or use x = e and t = 2 as limits in an expression involving ln (ln x)

Obtain correct solution in any form, e.g. ln (ln x) =− ln t +  ln 2

Use log laws to enable removal of logarithms

Obtain answer  $$x = e^{\frac{2}{t}}$$ , or simplified equivalent

(c) State that x tends to 1 coming from $$x = e^{\frac{k}{t}}$$

### Question

The variables x and y satisfy the differential equation
$$(1-cosx)\frac{dy}{dx}=y sin x$$.
It is given that y=4 when $$x=\pi$$.
(a) Solve the differential equation, obtaining an expression for y in terms of x.
(b) Sketch the graph of y against x for $$0<x<2\pi$$

Ans:

1. Separate variables correctly and attempt integration of at least one side
Obtain term In y
Obtain term of the form ±In(1-cosx)
Obtain term In(1-cosx)
Use $$x=\pi,y=4$$ to evaluate a constant, or as limits, in a solution containing terms of the form aIn y and bIn(1-cos x)
Obtain final answer y = 2(1-cosx)
2. Show a correct graph for $$0<x<2\pi$$ with the maximum at $$x=\pi$$

### Question

A large field of area 4 km2 is becoming infected with a soil disease. At time t years the area infected is x km2 and the rate of growth of the infected area is given by the differential equation dx
dt= kx 4 − x,
where k is a positive constant. It is given that when t = 0, x = 0.4 and that when t = 2, x = 2.
(i) Solve the differential equation and show that k = 1/4ln 3.
(ii) Find the value of t when 90% of the area of the field is infected.

### Question

The variables x and y satisfy the differential equation

$$x\frac{dy}{dx}=y(1-2x^{2})$$

and it is given that y = 2 when x = 1. Solve the differential equation and obtain an expression for y in
terms of x in a form not involving logarithms.

Separate variables and attempt integration of at least one side

Obtain term ln y

Obtain terms

Use x = 1 and y =2 to evaluate a constant, or as limits

Obtain correct solution in any form, e.g. 2 ln ln

Obtain correct expression for y, free of logarithms, i.e. $$y= 2x exp(1-x^{2})$$

### Question

The variables x and y satisfy the differential equation

$$\left ( x + 1 \right ) \left ( 3x + 1 \right )\frac{dy}{dx}= y,$$

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.

Ans:

Correctly separate variables and integrate at least one side

Obtain term ln y from integral of 1/y

State or imply the form  $$\frac{A}{x+1} + \frac{B}{3x+1}$$ and use a correct method to find a constant

Obtain $$A = -\frac{1}{2} and B = \frac{3}{2}$$

Obtain terms or combination of these terms

Use x = 1 and y = 1 to evaluate a constant, or expression for a constant, (decimal equivalent of ln terms allowed) or as limits, in a solution containing terms a ln y, b ln (x+1) and c ln (3x+1), where abc ≠ 0

Obtain an expression for y or y2 and substitute x = 3

Obtain answer y = $$\frac{1}{2}\sqrt{5} or \sqrt{\frac{5}{4}} or \sqrt{\frac{10}{8}}$$

### Question

The variables x and y are related by the differential equation

$$x\frac{dy}{dx}=1-y^{2}$$

When x = 2, y = 0. Solve the differential equation, obtaining an expression for y in terms of x.

Separate variables correctly and attempt integration of one side
Obtain term ln x

State or imply$$\frac{1}{1-y^{2}}\equiv \frac{A}{1-y}+\frac{B}{1+y}$$ and use a relevant method to find A or B

Obtain A$$=\frac{1}{2},B$$=$$\frac{1}{2}$$

Integrate and obtain$$-\frac{1}{2} In(1-y)+\frac{1}{2}In(1+y)$$,or equivalent

[If the integral is directly stated as$$k_{1}$$ In(\frac{1+y}{1-y}) \)or $$K_{2}$$In(\frac{1-y}{1+y})\) give , and then A2 for $$k_{1} or k_{2}$$=$$-\frac{1}{2}$$

Evaluate a constant, or use limits x = 2, y = 0 in a solution containing terms a ln x, b ln (1 – y) and c ln (1 + y), where$$abc\neq 0$$
[This M mark is not available if the integral of $$1/(1 – y^{2})$$is initially taken to be of the form k ln $$(1 – y^{2})$$]

Obtain solution in any correct form, e.g.$$\frac{1}{2} In(\frac{1+y}{1-y})$$ = Inx-In2

Rearrange and obtain  y=$$\frac{x^{2}-4}{x^{2}+4}$$ ,or equivalent, free of logarithms

### Question

The variables x and y are related by the differential equation $$\frac{dy}{dx}=\frac{6xe^{3x}}{y^{2}}$$ 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

.

Separate variables correctly and attempt integration on at least one side
Obtain $$\frac{1}{3}y^{3}$$
y or equivalent on left-hand side

Use integration by parts on right-hand side (as far as $$axe^{3x}+\int be^{3x}dx$$

Obtain or imply$$9 2xe^{3x}-\frac{2}{3}e^{3x}$$

Obtain
$$2xe^{3x}-\frac{2}{3}e^{3x}$$

Substitute x = 0, y = 2 in an expression containing terms where ABC ≠ 0, and
find the value of c

Obtain$$\frac{1}{3}y^{3}=2xe^{3x}-\frac{2}{3}e^{3x}+\frac{10}{3}$$ or equivalent

Substitute x = 0.5 to obtain y = 2.44

### Question

The variables x and y satisfy the differential equation

$$\frac{dy}{dx}=xe^{x+y},$$

and it is given that y = 0 when x = 0.

(i) Solve the differential equation and obtain an expression for y in terms of x. [7]

(ii) Explain briefly why x can only take values less than 1. [1]

Ans:

(i) Separate variables and attempt integration of one side
Obtain term -e-y
Integrate xex by parts reaching $$xe^{x}+\int e^{x} dx$$
Obtain integral xex -ex
Evaluate a constant, or use limits x = 0, y = 0
Obtain correct solution in any form
Obtain final answer y=-ln(ex(1-x)), or equivalent

(ii) Justify the given statement B1 [1]

### Question

Liquid is flowing into a small tank which has a leak. Initially the tank is empty and, t minutes later, the volume of liquid in the tank is V cm3. The liquid is flowing into the tank at a constant rate of 80 cm3 per minute. Because of the leak, liquid is being lost from the tank at a rate which, at any instant, is equal to kV cm3 per minute where k is a positive constant.

(i) Write down a differential equation describing this situation and solve it to show that [7]

$$V=\frac{1}{k}\left ( 80-80e^{-kt} \right )$$

(ii) It is observed that V = 500 when t = 15, so that k satisfies the equation

$$k=\frac{4-4e^{-kt} }{25}$$

Use an iterative formula, based on this equation, to find the value of k correct to 2 significant figures. Use an initial value of k = 0.1 and show the result of each iteration to 4 significant figures. [3]

(iii) Determine how much liquid there is in the tank 20 minutes after the liquid started flowing, and state what happens to the volume of liquid in the tank after a long time.[2]

Ans:

(i) State $$\frac{dV}{dt}=80-kV$$
Correctly separate variables and attempt integration of one side
Obtain a ln(80 − kV) = t or equivalent
Obtain $$-\frac{1}{k}ln\left ( 80-kV \right )=t$$ or equivalent
Use t = 0 and V = 0 to find constant of integration or as limits
Obtain $$-\frac{1}{k}ln\left ( 80-kV \right )=t-\frac{1}{k}$$ or equivalent
Obtain given answer $$V=\frac{1}{k}\left ( 80-80e^{-kt} \right )$$ correctly

(ii) Use iterative formula correctly at least once
Obtain final answer 0.14
Show sufficient iterations to 4 s.f. to justify answer to 2 s.f. or show a sign change in the interval (0.135, 0.145)

(iii) State a value between 530 and 540 cm3 inclusive
State or imply that volume approaches 569 cm3 (allowing any value between 567 and 571 inclusive)

### Question

The variables x and y satisfy the differential equation $$\frac{dy}{dx}=4cos^{2}ytanx.for 0\leqslant x< \frac{1}{2}\pi , and x=0 when y=\frac{1}{4}\pi$$,Solve this differential equation and find the value of x when $$\frac{1}{3}\pi$$

Separate variables correctly and attempt integration of one side  Obtain term tan y , or equivalent
Obtain term of the form k x ln cos , or equivalent
Obtain term −4ln cos x , or equivalent  Use x = 0 and  y =$$\frac{1}{4} π$$ in solution containing a y tan and b x ln cos to evaluate a constant, or as limits Obtain correct solution in any form, e.g. tan 4ln secx+1  Substitute y =$$\frac{1}{3} π$$  in solution containing terms a y tan and b x ln cos , and use correct
method to find x Obtain answer x = 0.587

### Question

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

$$x\frac{dy}{dx}=(1-2x^{2})y,$$

for x > 0. It is given that y = 1 when x = 1.
Solve the differential equation, obtaining an expression for y in terms of x.                                             [6]

Separate variables correctly and attempt integration of at least
one side
Obtain term ln y
Obtain terms 2 ln x − x2
Use x = 1, y = 1 to evaluate a constant, or as limits, in a solution
containing at least 2 terms of the form a ln  y , b  ln x and cx
Obtain correct solution in any form
Rearrange and obtain $$y=xe^{1-x^{2}}$$

### Question

A certain curve is such that its gradient at a point (x, y) is proportional to $$\frac{y}{x\sqrt{x}}$$. 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.                                                                                                                                                                          [8]

(b) Describe what happens to y as x tends to infinity.                                                                                                     [1]

Ans

(a) State $$\frac{dy}{dx}=k\frac{y}{x\sqrt{x}}$$, or equivalent

Separate variables correctly and attempt integration of at least one side M1
Obtain term ln y, or equivalent
Obtain term $$-2k\frac{1}{\sqrt{x}}$$, or equivalent

Use given coordinates to find k or a constant of integration c in a solution containing terms of the form a ln y and $$\frac{b}{\sqrt{x}}$$,  where ab≠ 0
Obtain k = 1 and c = 2

Obtain final answer y = exp $$\left ( -\frac{2}{\sqrt{x}}+2 \right )$$, or equivalent

(b) State that y approaches e2
(FT their c in part (a) of the correct form)

### Question

The variables x and y satisfy the differential equation
$$\frac{dy}{dx}=\frac{1+4y^2}{e^x}$$.
It is given that y=0 when x = 1.
(a) Solve the differential equation, obtaining an expression for y in terms of x.
(b) State what happens to the value of y as x tends to infinity.

Ans:

(a) Separate variables correctly and attempt integration of at least one side
Obtain term of the form $$atan^{-1}(2y)$$
Obtain term $$\frac{1}{2}tan^{-1}(2y)$$
Obtain term $$-e^{-x}$$
Use x = 1, y = 0 to evaluate a constant or as limits in a solution containing
terms of the form $$atan^{-1}(by)$$ and $$ce^{\pm x}$$
Obtain correct answer in any form
Obtain final answer $$y=\frac{1}{2}tan(2e^{-1}-2e^{-x})$$, or equivalent
(b) State that y approaches $$\frac{1}{2}tan(2e^{-1})$$, or equivalent

### Question

(a)Given that y = ln ( ln x), show that

$$\frac{dy}{dx} = \frac{1}{x ln x}.$$

The variables x and t satisfy the differential equation

x ln x + t $$\frac{dx}{dt} = 0$$

It is given that x = e when t = 2.

(b)Solve the differential equation obtaining an expression for x in terms of t, simplifying your answer.

(c)Hence state what happens to the value of x as t tends to infinity.

Ans:

(a)Show sufficient working to justify the given answer

(b)Correct separation of variables

Obtain term ln  ( ln x)

Obtain term −ln t

Evaluate a constant or use x = e and t = 2 as limits in an expression involving ln (ln x)

Obtain correct solution in any form, e.g. ln (ln x) =− ln t +  ln 2

Use log laws to enable removal of logarithms

Obtain answer  $$x = e^{\frac{2}{t}}$$ , or simplified equivalent

(c) State that x tends to 1 coming from $$x = e^{\frac{k}{t}}$$

### Question

The variables x and y satisfy the differential equation
$$(1-cosx)\frac{dy}{dx}=y sin x$$.
It is given that y=4 when $$x=\pi$$.
(a) Solve the differential equation, obtaining an expression for y in terms of x.
(b) Sketch the graph of y against x for $$0<x<2\pi$$

Ans:

1. Separate variables correctly and attempt integration of at least one side
Obtain term In y
Obtain term of the form ±In(1-cosx)
Obtain term In(1-cosx)
Use $$x=\pi,y=4$$ to evaluate a constant, or as limits, in a solution containing terms of the form aIn y and bIn(1-cos x)
Obtain final answer y = 2(1-cosx)
2. Show a correct graph for $$0<x<2\pi$$ with the maximum at $$x=\pi$$

### Question

A large field of area 4 km2 is becoming infected with a soil disease. At time t years the area infected is x km2 and the rate of growth of the infected area is given by the differential equation dx
dt= kx 4 − x,
where k is a positive constant. It is given that when t = 0, x = 0.4 and that when t = 2, x = 2.
(i) Solve the differential equation and show that k = 1/4ln 3.
(ii) Find the value of t when 90% of the area of the field is infected.

### Question

The variables x and y satisfy the differential equation

$$x\frac{dy}{dx}=y(1-2x^{2})$$

and it is given that y = 2 when x = 1. Solve the differential equation and obtain an expression for y in
terms of x in a form not involving logarithms.

Separate variables and attempt integration of at least one side

Obtain term ln y

Obtain terms

Use x = 1 and y =2 to evaluate a constant, or as limits

Obtain correct solution in any form, e.g. 2 ln ln

Obtain correct expression for y, free of logarithms, i.e. $$y= 2x exp(1-x^{2})$$

### Question

The variables x and y satisfy the differential equation

$$\left ( x + 1 \right ) \left ( 3x + 1 \right )\frac{dy}{dx}= y,$$

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.

Ans:

Correctly separate variables and integrate at least one side

Obtain term ln y from integral of 1/y

State or imply the form  $$\frac{A}{x+1} + \frac{B}{3x+1}$$ and use a correct method to find a constant

Obtain $$A = -\frac{1}{2} and B = \frac{3}{2}$$

Obtain terms or combination of these terms

Use x = 1 and y = 1 to evaluate a constant, or expression for a constant, (decimal equivalent of ln terms allowed) or as limits, in a solution containing terms a ln y, b ln (x+1) and c ln (3x+1), where abc ≠ 0

Obtain an expression for y or y2 and substitute x = 3

Obtain answer y = $$\frac{1}{2}\sqrt{5} or \sqrt{\frac{5}{4}} or \sqrt{\frac{10}{8}}$$

### Question

The variables x and y are related by the differential equation

$$x\frac{dy}{dx}=1-y^{2}$$

When x = 2, y = 0. Solve the differential equation, obtaining an expression for y in terms of x.

Separate variables correctly and attempt integration of one side
Obtain term ln x

State or imply$$\frac{1}{1-y^{2}}\equiv \frac{A}{1-y}+\frac{B}{1+y}$$ and use a relevant method to find A or B

Obtain A$$=\frac{1}{2},B$$=$$\frac{1}{2}$$

Integrate and obtain$$-\frac{1}{2} In(1-y)+\frac{1}{2}In(1+y)$$,or equivalent

[If the integral is directly stated as$$k_{1}$$ In(\frac{1+y}{1-y}) \)or $$K_{2}$$In(\frac{1-y}{1+y})\) give , and then A2 for $$k_{1} or k_{2}$$=$$-\frac{1}{2}$$

Evaluate a constant, or use limits x = 2, y = 0 in a solution containing terms a ln x, b ln (1 – y) and c ln (1 + y), where$$abc\neq 0$$
[This M mark is not available if the integral of $$1/(1 – y^{2})$$is initially taken to be of the form k ln $$(1 – y^{2})$$]

Obtain solution in any correct form, e.g.$$\frac{1}{2} In(\frac{1+y}{1-y})$$ = Inx-In2

Rearrange and obtain  y=$$\frac{x^{2}-4}{x^{2}+4}$$ ,or equivalent, free of logarithms

### Question

The variables x and y are related by the differential equation $$\frac{dy}{dx}=\frac{6xe^{3x}}{y^{2}}$$ 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

.

Separate variables correctly and attempt integration on at least one side
Obtain $$\frac{1}{3}y^{3}$$
y or equivalent on left-hand side

Use integration by parts on right-hand side (as far as $$axe^{3x}+\int be^{3x}dx$$

Obtain or imply$$9 2xe^{3x}-\frac{2}{3}e^{3x}$$

Obtain
$$2xe^{3x}-\frac{2}{3}e^{3x}$$

Substitute x = 0, y = 2 in an expression containing terms where ABC ≠ 0, and
find the value of c

Obtain$$\frac{1}{3}y^{3}=2xe^{3x}-\frac{2}{3}e^{3x}+\frac{10}{3}$$ or equivalent

Substitute x = 0.5 to obtain y = 2.44

### Question

The variables x and y satisfy the differential equation

$$\frac{dy}{dx}=xe^{x+y},$$

and it is given that y = 0 when x = 0.

(i) Solve the differential equation and obtain an expression for y in terms of x. [7]

(ii) Explain briefly why x can only take values less than 1. [1]

Ans:

(i) Separate variables and attempt integration of one side
Obtain term -e-y
Integrate xex by parts reaching $$xe^{x}+\int e^{x} dx$$
Obtain integral xex -ex
Evaluate a constant, or use limits x = 0, y = 0
Obtain correct solution in any form
Obtain final answer y=-ln(ex(1-x)), or equivalent

(ii) Justify the given statement B1 [1]

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