**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]

**Answer/Explanation**

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.

**Answer/Explanation**

**(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 \(\Theta \)

**Answer/Explanation**

**(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**

**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.

**Answer/Explanation**

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 \(V=\frac{1}{3}\pi r^{2}h.\) ]

**Answer/Explanation**

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 cm^{3}. The liquid is flowing into the tank at a constant rate of 80 cm^{3 }per minute. Because of the leak, liquid is being lost from the tank at a rate which, at any instant, is equal to kV cm^{3 }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]

**Answer/Explanation**

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 cm^{3} inclusive

State or imply that volume approaches 569 cm^{3} (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 \(\frac{1}{3}\pi \)

**Answer/Explanation**

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]

**Answer/Explanation**

Separate variables correctly and attempt integration of at least

one side

Obtain term ln y

Obtain terms 2 ln x − x^{2}

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^{2 }

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]

**Answer/Explanation**

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 e^{2}

(FT *their c* in part (a) of the correct form)

**Question**

**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.

**Answer/Explanation**

**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.

**Answer/Explanation**

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**

**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 \(0<x<2\pi\)

**Answer/Explanation**

**Ans:**

- 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) - Show a correct graph for \(0<x<2\pi\) with the maximum at \(x=\pi\)

**Question**

**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.

**Answer/Explanation**

**Question**

**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.

**Answer/Explanation**

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.

**Answer/Explanation**

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 y^{2} and substitute x = 3

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

**Question**

**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

**Answer/Explanation**

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**

**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

**Answer/Explanation**

.

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]

**Answer/Explanation**

Ans:

** (i)** Separate variables and attempt integration of one side

Obtain term -e^{-y}

Integrate *x*e^{x} by parts reaching \(xe^{x}+\int e^{x} dx\)

Obtain integral *x*e^{x }*-e ^{x} *

Evaluate a constant, or use limits

*x*= 0,

*y*= 0

Obtain correct solution in any form

Obtain final answer

*y*=-ln(e

^{x}(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 cm^{3}. The liquid is flowing into the tank at a constant rate of 80 cm^{3 }per minute. Because of the leak, liquid is being lost from the tank at a rate which, at any instant, is equal to kV cm^{3 }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]

**Answer/Explanation**

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 cm^{3} inclusive

State or imply that volume approaches 569 cm^{3} (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 \(\frac{1}{3}\pi \)

**Answer/Explanation**

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]

**Answer/Explanation**

Separate variables correctly and attempt integration of at least

one side

Obtain term ln y

Obtain terms 2 ln x − x^{2}

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^{2 }

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]

**Answer/Explanation**

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 e^{2}

(FT *their c* in part (a) of the correct form)

**Question**

**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.

**Answer/Explanation**

**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.

**Answer/Explanation**

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**

**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 \(0<x<2\pi\)

**Answer/Explanation**

**Ans:**

- 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) - Show a correct graph for \(0<x<2\pi\) with the maximum at \(x=\pi\)

**Question**

**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.

**Answer/Explanation**

**Question**

**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.

**Answer/Explanation**

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.

**Answer/Explanation**

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 y^{2} and substitute x = 3

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

**Question**

**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

**Answer/Explanation**

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**

**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

**Answer/Explanation**

.

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]

**Answer/Explanation**

Ans:

** (i)** Separate variables and attempt integration of one side

Obtain term -e^{-y}

Integrate *x*e^{x} by parts reaching \(xe^{x}+\int e^{x} dx\)

Obtain integral *x*e^{x }*-e ^{x} *

Evaluate a constant, or use limits

*x*= 0,

*y*= 0

Obtain correct solution in any form

Obtain final answer

*y*=-ln(e

^{x}(1-

*x*)), or equivalent

**(ii)** Justify the given statement B1 [1]