DP IB Maths: AI HL

Topic Questions

5.6 Differential Equations

1
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5 marks

Consider the first-order differential equation

 fraction numerator d y over denominator d x end fraction minus x cubed equals 2 sin x 

Solve the equation given that y equals 0 when x equals 0,  giving your answer in the form y equals f open parentheses x close parentheses.

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2a
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4 marks

Use separation of variables to solve each of the following differential equations:

fraction numerator d y over denominator d x end fraction equals 10 x cubed y cubed

2b
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5 marks

fraction numerator d y over denominator d x end fraction equals x open parentheses x squared minus 1 close parentheses cubed e to the power of 3 y end exponent

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3a
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5 marks

Use separation of variables to solve each of the following differential equations for y which satisfies the given boundary condition:

fraction numerator d y over denominator d x end fraction equals fraction numerator cos 3 x over denominator y end fraction semicolon space space space y open parentheses straight pi over 6 close parentheses equals negative 1

3b
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5 marks

e to the power of 2 x end exponent fraction numerator d y over denominator d x end fraction equals cos squared y semicolon space space space space y open parentheses 0 close parentheses equals straight pi over 4

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4a
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5 marks

Radiangast the Beige is chief mathemagician of the wizards’ council.  After animals begin falling ill in the forest where he lives, Radiangast realises that an evil magic has begun spreading through the forest.  After studying the situation, he believes that at any point in time, t, the rate of change of the area, A, affected by the evil magic is inversely proportional to the square root of the area already affected. 

Write down a differential equation representing Radiangast’s model, and solve it to find the general solution.  Be sure to define any constants that occur in your equation or solution.

4b
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6 marks

At the time when Radiangast first noticed its presence, the evil magic was affecting an area of 16 acres of forest.  One week later he noticed that the area has increased to 41 acres.

Radiangast knows that as long as the wizards’ council convenes to weave spells before the area affected by the evil magic exceeds 100 acres, then they will be able to stop the evil magic from spreading further. 

From the time that Radiangast first noticed the presence of the evil magic, determine how long the wizards’ council has to convene to weave spells, if they are to stop the evil magic from spreading further.

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5a
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8 marks

After an invasive species of insect has been introduced to a new region, it is estimated that at any point in time  the rate of growth of the population of insects in the region will be proportional to the current population size P. At the start of a study of the insects in a particular region, researchers estimate the population size to be 1000 individuals. A week later another population survey is conducted, and the population of insects is found to have increased to 1150.

By first writing and solving an appropriate differential equation, determine how long it will take for the population of insects in the region to increase to 10 000.

5b
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2 marks

Comment on the validity of the model for large values of t.

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6a
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3 marks

The graph below shows the slope field for the differential equation fraction numerator d y over denominator d x end fraction equals fraction numerator open parentheses 0.2 x minus 0.85 close parentheses y over denominator open parentheses 0.75 minus 0.2 y close parentheses x end fraction comma x greater than 0 comma space y greater than 0 comma in the intervals 0 less than x less or equal than 10 and 0 less than y less or equal than 10.

mi_q6a_5-6_differential-equations_hard_ib_ai_hl_maths_dig

Find the equations of the lines on which will lie the points where the solution curves to the differential equation have (i) horizontal and (ii) vertical tangents.

6b
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5 marks

On the graph above sketch:

i)
the lines identified in part (a)
ii)
the solution curve that passes through the point (8, 6)
iii)
the solution curve that passes through the point (4, 6)

 

 

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7a
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3 marks

Consider the differential equation

 fraction numerator d y over denominator d x end fraction equals open parentheses fraction numerator 1 over denominator e to the power of square root of x end exponent cos space x end fraction close parentheses squared minus fraction numerator y over denominator square root of x end fraction 

with the boundary condition y open parentheses straight pi over 3 close parentheses equals 0

Apply Euler’s method with a step size of h equals 0.01 to approximate the solution to the differential equation at x equals fraction numerator 20 straight pi plus 3 over denominator 60 end fraction.

7b
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7 marks

It can be shown that the exact solution to the differential equation with the given boundary condition is

y equals fraction numerator tan space x minus square root of 3 over denominator e to the power of square root of x end exponent end fraction 

(i)
Compare your approximation from part (a) to the exact value of the solution at x equals fraction numerator 20 straight pi plus 3 over denominator 60 end fraction.

(ii)
Explain how the accuracy of the approximation in part (a) could be improved.

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8a
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3 marks

A particle moves in a straight line, such that its displacement x at time t is described by the differential equation

x equals fraction numerator t space sin space t squared over denominator cos space x end fraction comma space space space space space space space t greater or equal than 0 

At time t equals 0 comma space x equals negative straight pi over 3 

By using Euler’s method with a step length of 0.2, find an approximate value for x at time t equals 0.6.

8b
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5 marks

Solve the differential equation with the given boundary condition.

8c
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2 marks

Hence find the percentage error in your approximation for x at time  t equals 0.6.

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1a
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3 marks

Consider the first-order differential equation

 fraction numerator d y over denominator d x end fraction minus 5 x to the power of 4 equals 3 

Find the general solution to the differential equation, giving your answer in the form y equals f left parenthesis x right parenthesis.

1b
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2 marks

Find the specific solution to the equation given that y equals 40 space spacewhen  x equals 2.

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2a
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4 marks

Use separation of variables to find the general solution of each of the following differential equations, giving your answers in the form y equals f left parenthesis x right parenthesis :

fraction numerator d y over denominator d x end fraction equals fraction numerator 4 x squared over denominator y to the power of 4 end fraction

 

2b
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4 marks

fraction numerator d y over denominator d x end fraction equals left parenthesis x squared plus 1 right parenthesis e to the power of negative y end exponent

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3a
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5 marks

Use separation of variables to solve each of the following differential equations for y which satisfies the given boundary condition:   

            fraction numerator straight d y over denominator straight d x end fraction equals x y squared semicolon blank y open parentheses 2 close parentheses equals 1

3b
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5 marks
            open parentheses x plus 3 close parentheses fraction numerator straight d y over denominator straight d x end fraction equals sec space y semicolon blank y open parentheses negative 2 close parentheses equals fraction numerator 3 pi over denominator 2 end fraction

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4a
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2 marks

Scientists are studying a large pond where an invasive plant has been observed growing, and they have begun measuring the area, A straight m squared comma of the pond’s surface that is covered by the plant.  According to the scientists’ model, the rate of change of the area of the pond covered by the plant at any time, t, is proportional to the square root of the area already covered.

Write down a differential equation to represent the scientists’ model.

4b
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4 marks

Solve the differential equation to show that

A equals open parentheses fraction numerator k t plus c over denominator 2 end fraction close parentheses squared

 where k is the constant of proportionality and c is a constant of integration

4c
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3 marks

At the time when the scientists begin studying the pond the invasive plant covers an area of 100 m2 .  One week later the area has increased to 225 m2.

Use this information to determine the values of k and c.

 

4d
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2 marks

The pond has a total area of 250   000 space straight m squared .

Determine how long it will take, according to the scientists’ model, for the invasive plant to cover the entire surface of the pond.

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5a
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1 mark

At any point in time, the rate of growth of a colony of bacteria is proportional to the current population size P. At time t equals 0 hours, the population size is 5000.

Write a differential equation to model the size of the population of bacteria.

5b
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6 marks

After 1 hour, the population has grown to 7000.

By first solving the differential equation from part (a), determine the constant of proportionality.

5c
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5 marks
(i)
Show that, according to the model, it will take exactly fraction numerator ln space 20 over denominator ln space 7 minus ln space 5 end fraction  hours (from t equals 0)  for the population of bacteria to grow to 100 space 000.
(ii)
Confirm your answer to part (c)(i) graphically.

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6a
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1 mark

The graph below shows the slope field for the differential equation fraction numerator d y over denominator d x end fraction equals e to the power of x plus y   in the intervals negative 1 less or equal than x less or equal than 2 space space and negative 3 less or equal than y less or equal than 1.

q6-differential-equation-ib-ai-hl-maths-screenshot

Calculate the value of fraction numerator d y over denominator d x end fractionat the point left parenthesis 0 comma negative 3 right parenthesis.

6b
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6 marks

On the graph above sketch:

(i)
a curve that represents the points where fraction numerator d y over denominator d x end fraction equals 0
(ii)
the solution curve that passes through the point left parenthesis 0 comma negative 1 right parenthesis
(iii)
the solution curve that passes through the point left parenthesis 0 comma negative 2 right parenthesis

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7a
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3 marks

Consider the differential equation

 fraction numerator d y over denominator d x end fraction equals y over x plus 1 

with the boundary condition  y left parenthesis 1 right parenthesis equals 0..

Apply Euler’s method with a step size of h equals 0.2  to approximate the solution to the differential equation at x equals 2. .

7b
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3 marks

It can be shown that the exact solution to the differential equation with the given boundary condition is y equals x space ln space x.  Compare your approximation from part (a) to the exact value of the solution at x equals 2.

7c
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1 mark

Explain how the accuracy of the approximation in part (a) could be improved.

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8a
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3 marks

A particle moves in a straight line, such that its displacement x at time t is described by the differential equation.

x with dot on top equals fraction numerator t e to the power of 3 t squared end exponent plus 1 over denominator 4 x squared end fraction   comma space t greater or equal than 0

 At time  space t equals 0 comma space space x equals 1 half  .

By using Euler’s method with a step length of 0.1 , find an approximate value for x  at time t equals 0.3.

8b
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5 marks
(i)
Solve the differential equation with the given boundary condition to show that  
x equals 1 half cube root of e to the power of 3 t squared end exponent plus 6 t end root

 

(ii)
Hence find the percentage error in your approximation for x at time  t equals 0.3.

 

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1
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5 marks

Consider the first-order differential equation

 fraction numerator d y over denominator d x end fraction plus fraction numerator 1 over denominator 2 x end fraction equals sin space 3 x space cos space 3 x 

By first finding the general solution to the equation, solve the equation for the case that  y=0   when x equals straight pi over 2.

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2a
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4 marks

Use separation of variables to find the general solution of each of the following differential equations:

fraction numerator d y over denominator d x end fraction equals fraction numerator 3 y to the power of 4 over denominator 4 x cubed end fraction

2b
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5 marks

fraction numerator d y over denominator d x end fraction equals fraction numerator x squared over denominator y open parentheses straight pi minus x cubed close parentheses end fraction e to the power of y squared end exponent

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3a
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5 marks

Solve each of the following differential equations for y which satisfies the given boundary condition.

cos space straight pi x to the power of 4 fraction numerator d y over denominator d x end fraction equals open parentheses x over y close parentheses cubed tan space straight pi x to the power of 4 semicolon space space space space space space space y open parentheses 0 close parentheses equals negative 3

3b
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6 marks

open parentheses fraction numerator e to the power of x squared end exponent over denominator cos space y end fraction close parentheses space fraction numerator d y over denominator d x end fraction equals x squared space cos space y semicolon space space space space space space y open parentheses 0 close parentheses equals fraction numerator 3 straight pi over denominator 4 end fraction

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4
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10 marks

The evil Galactic Imperium has been spreading through the galaxy, taking over larger and larger volumes of galactic space as time goes on.  The area of space controlled by the Imperium at any point in time may be modelled as a sphere centred on the capital planet Merekhty.

Representatives of the Star Rebellion are on the planet Nezal, attempting to convince the planet’s inhabitants to join the rebellion.  Nezal lies 16.2 kiloparsecs (kpc) away from Merekhty, however, and because of that great distance the inhabitants of the planet believe it will be a very long time before they need to worry about the Imperium’s expansion.

As the Rebellion’s Chief Mathematician, you have been given the job of preparing a report on the expansion of the Imperium in relation to Nezal.  Based on your research, you believe that at any time, , the rate of expansion of the volume of space controlled by the Imperium, , is inversely proportional to the square of the cube root of the volume of space already controlled by the Imperium at that time.

Given that one year ago the Imperium controlled 8 cubic kiloparsecs of galactic space, whereas now it controls 2197 cubic kiloparsecs, determine how many more years it will be before Nezal falls within the Imperium’s sphere of control.

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5a
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8 marks

As the atoms in a sample of radioactive material undergo radioactive decay, the rate of change of the number of radioactive atoms remaining in the sample at any time t is proportional to the number, N, of radioactive atoms currently remaining.  The amount of time, lambda, that it takes for half the radioactive atoms in a sample of radioactive material to decay is known as the ­half-life of the material. 

Let N subscript 0 be the number of radioactive atoms originally present in a sample. 

By first writing and solving an appropriate differential equation, show that the number of radioactive atoms remaining in the sample at any time t greater or equal than 0 may be expressed as

N open parentheses t close parentheses equals N subscript 0 e to the power of negative fraction numerator ln space 2 over denominator lambda end fraction t end exponent
5b
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3 marks

Plutonium-239, a by-product of uranium fission reactors, has a half-life of 24000 years.

For a particular sample of Plutonium-239, determine how long it will take until less than 1% of the original radioactive Plutonium-239 atoms in the sample remain.

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6a
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2 marks

The diagram below shows the slope field for the differential equation

fraction numerator d y over denominator d x end fraction equals cos open parentheses x minus y close parentheses 

The graphs of the two solutions to the differential equation that pass through the points open parentheses 0 comma straight pi over 3 close parentheses and open parentheses 0 comma straight pi close parentheses are shown.

mi_q6a_5-6_differential-equations_very_hard_ib_ai_hl_maths_dig

Explain the relationship that must exist between x and y for fraction numerator d y over denominator d x end fraction=0 to be true.

 

6b
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3 marks

For the two solutions given, the local minimum points lie on the straight line L subscript 1and the local maximum points lie on the straight line L subscript 2

Find the equations of  (i) L subscript 1 and  (ii) L subscript 2,  giving your answers in the form y equals m x plus c.

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7a
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3 marks

Consider the differential equation

 fraction numerator d y over denominator d x end fraction equals fraction numerator 5 over denominator square root of 63 plus 11 x squared minus 2 x to the power of 4 end root end fraction minus fraction numerator 2 x y over denominator 2 x squared plus 7 end fraction 

with the boundary condition y open parentheses negative fraction numerator 3 square root of 2 over denominator 2 end fraction close parentheses equals 1.

Apply Euler’s method with a step size of  h equals 0.2 to approximate the solution to the differential equation at x equals fraction numerator 2 minus 3 square root of 3 over denominator 2 end fraction .

7b
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3 marks

It can be shown that exact solution to the differential equation with the given boundary condition is

y equals fraction numerator 5 sin to the power of negative 1 end exponent open parentheses x over 3 close parentheses plus 4 plus fraction numerator 5 straight pi over denominator 4 end fraction over denominator square root of 2 x squared plus 7 end root end fraction

(i)
Compare your approximation from part (a) to the exact value of the solution at x equals fraction numerator 2 minus 3 square root of 2 over denominator 2 end fraction.
(ii)
Explain how the accuracy of the approximation in part (a) could be improved.

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8a
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3 marks

A particle moves in a straight line, such that its displacement x at time t is described by the differential equation

 x equals fraction numerator t open parentheses t squared plus 1 close parentheses e to the power of negative 3 x end exponent over denominator open parentheses 2 t squared plus 1 close parentheses open parentheses 2 t squared plus 3 close parentheses end fraction comma space space space space space space space space space space t greater or equal than 0

At time t equals 0x equals 0.

By using Euler’s method with a step length of 0.1, find an approximate value for x at time t equals 0.3.

8b
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5 marks

Solve the differential equation with the given boundary condition.

8c
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2 marks

Hence find the percentage error in your approximation for x at time t equals 0.3.

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