# 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.                                                                                                                                                                                                                        

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.

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

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

[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 

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

(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. 

(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.

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
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.                                             

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.                                                                                                                                                                          

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

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. 

(ii) Explain briefly why x can only take values less than 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 

### 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 

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

(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. 

(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.

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
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.                                             

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.                                                                                                                                                                          

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

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. 

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

Integrate xex by parts reaching $$xe^{x}+\int e^{x} dx$$