IBMaths: AA HLDPTopic Questions5. Calculus5.10 Differential Equations

Differential Equations (DP IB Maths: AA HL)

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5.10 Differential Equations

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1
Sme Calculator5 marks

Consider the first-order differential equation

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

Solve the equation given that y equals 40  when x equals 2,  giving your answer in the form y equals f left parenthesis x right parenthesis.

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2a
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Use separation of variables to solve each of the following differential equations for y:

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

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2b
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fraction numerator straight d y over denominator straight d x end fraction equals open parentheses x squared plus 1 close parentheses e to the power of negative y end exponent

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3a1 mark

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

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3b
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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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At any point in time, the rate of growth of a colony of bacteria is proportional to the current population size. At time t equals 0 hours, the population size is 5000.

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

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4b
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After 1 hour, the population has grown to 7000.

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

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4c
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(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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5a
Sme Calculator8 marks

After clearing a large forest of malign influences, a wizard introduces a population of 100 unicorns to the forest.  According to the wizard’s mathemagicians, the population of unicorns in the forest may be modelled by the logistic equation

 fraction numerator straight d P over denominator straight d t end fraction equals 0.0006   P open parentheses 250 minus P close parentheses

where t is the time in years after the unicorns were introduced to the forest.

Show that the population of unicorns at time t years is given by  

P open parentheses t close parentheses equals fraction numerator 500 straight e to the power of 0.15 t end exponent over denominator 3 plus 2 straight e to the power of 0.15 straight t end exponent end fraction

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5b
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Find the length of time predicted by the model for the population of unicorns to double in size.

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5c
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Determine the maximum size that the model predicts the population of unicorns can grow to.

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6a
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Show that

 x squared fraction numerator straight d y over denominator straight d x end fraction equals x y plus 2 x squared

is a homogeneous differential equation.

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6b
Sme Calculator4 marks

Using the substitution v equals y over x,  show that the solution to the differential equation in part (a) is

 y equals 2 x ln open vertical bar x close vertical bar plus c x 

where c is a constant of integration.

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7a
Sme Calculator3 marks

Use the substitution v equals y over x  to show that the differential equation 

y to the power of apostrophe equals y squared over x squared minus y over x plus 1

may be rewritten in the form

v to the power of apostrophe equals open parentheses v minus 1 close parentheses squared over x

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7b
Sme Calculator5 marks

Hence use separation of variables to solve the differential equation in part (a) for which satisfies the boundary condition begin mathsize 16px style y open parentheses 1 close parentheses equals 2 over 3 end style. Give your answer in the form y equals f left parenthesis x right parenthesis.

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8a
Sme Calculator2 marks

Consider the differential equation

 y to the power of apostrophe plus 2 x y equals left parenthesis 4 x plus 2 right parenthesis e to the power of x

Explain why it would be appropriate to use an integrating factor in attempting to solve the differential equation.

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8b
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Show that the integrating factor for this differential equation is straight e to the power of x squared end exponent.

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8c
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Hence solve the differential equation.

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9
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Use an integrating factor to solve the differential equation

 open parentheses x plus 3 close parentheses fraction numerator straight d y over denominator straight d x end fraction minus 4 y equals open parentheses x plus 3 close parentheses to the power of 6 

for y which satisfies the boundary condition  y left parenthesis negative 2 right parenthesis equals 0.

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10a
Sme Calculator3 marks

Consider the differential equation

 fraction numerator straight d y over denominator straight 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.

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10b
Sme Calculator4 marks
(i)
Explain what method you could use to solve the above differential equation analytically (i.e., exactly).

(ii)
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.

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10c
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Explain how the accuracy of the approximation in part (a) could be improved.

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11a
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A particle moves in a straight line, such that its displacement x at time t is described by the differential equation

fraction numerator straight d x over denominator straight d t end fraction 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 space space space space t greater or equal than 0  

At time t equals 0 , 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.

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11b
Sme Calculator5 marks
(i)
Solve the differential equation with the given boundary condition to show that 

 x equals 1 half s-th 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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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
Sme Calculator4 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

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2b
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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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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

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3b
Sme Calculator5 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
Sme Calculator8 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.

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4b
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Comment on the validity of the model for large values of t.

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5a
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Ignoring the advice of her father’s professional dragon keepers, Princess Sarff releases her personal menagerie of 800 dragons onto the archipelago known as the Sheep Islands. Sarff believes that the dragons will thrive in such a sheep-rich environment. The chief dragon keeper, however, has studied the sheep population of the islands as well as the appetite of dragons. Based on his research, he believes that the population P of dragons in the islands may be modelled by the logistic equation

fraction numerator d P over denominator d t end fraction equals 0.00025 P open parentheses 160 minus P close parentheses

where t is the time in years after the dragons were introduced to the archipelago. 

Use the logistic equation to explain why, according to the model, the dragon population will initially be decreasing.

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5b
Sme Calculator10 marks

By first solving the logistic equation for P, determine the amount of time it will take for the dragon population to shrink to half its original size.

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5c
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Determine the long-term trend for the dragon population, using mathematical reasoning to justify your answer.

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6a
Sme Calculator2 marks

Consider the differential equation

 open parentheses x squared plus y squared close parentheses fraction numerator d y over denominator d x end fraction equals x y 

Explain why the substitution v equals y over x would be an appropriate method to use to solve the differential equation.

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6b
Sme Calculator5 marks

Show that the solution to the differential equation may be expressed in the form

y equals A e to the power of fraction numerator x squared over denominator 2 y squared end fraction end exponent

where A is an arbitrary constant.

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6c
Sme Calculator3 marks

Find the precise solution to the differential equation given that y equals 1 half when x equals 1.

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7
Sme Calculator8 marks

Use the substitution v equals y over x to solve the differential equation

x squared y apostrophe equals y squared plus 7 x y plus 9 x squared 

for y which satisfies the boundary condition y open parentheses 1 close parentheses equals negative 2. Give your answer in the form y equals f open parentheses x close parentheses .

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8
Sme Calculator6 marks

Use an integrating factor to solve the differential equation

x y apostrophe plus 2 y equals 1 plus e to the power of x squared end exponent

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9a
Sme Calculator3 marks

Consider the differential equation

 fraction numerator d y over denominator d x end fraction equals open parentheses fraction numerator sec space x over denominator e to the power of square root of x end exponent 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.

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9b
Sme Calculator7 marks

Solve the differential equation analytically, for y which satisfies the given boundary condition.

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9c
Sme Calculator3 marks
(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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10a
Sme Calculator3 marks

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

 fraction numerator d x over denominator d t end fraction equals fraction numerator 1 over denominator 1 plus sin open parentheses t plus 1 close parentheses minus cos open parentheses t plus 1 close parentheses end fraction space comma space space space space space space space space 0 less or equal than t less or equal than 3.5 

At time t equals 2 , x equals 1.

By using Euler’s method with a step length of 0.25, find an approximate value for x at time t equals 3.25.

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10b
Sme Calculator3 marks

The diagram below shows a graph of the exact solution  x equals f open parentheses t close parentheses to the differential equation with the given boundary condition.

q10b_5-10_differential-equations_hard_ib_aa_hl_maths_diagram

Explain using the graph whether the approximation found in part (a) will be an overestimate or an underestimate for the true value of x when t equals 3.25.  Be sure to use mathematical reasoning to justify your answer.

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1
Sme Calculator5 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 

Solve the equation given that y equals 0 when  x equals straight pi over 2,  giving your answer in the form  y equals f left parenthesis x right parenthesis.       .

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2a
Sme Calculator4 marks

Use separation of variables to solve 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

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2b
Sme Calculator5 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
Sme Calculator5 marks

Solve each of the following differential equations for y which satisfies the given boundary condition, giving your answers in the form y equals f open parentheses x close parentheses.

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

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3b
Sme Calculator6 marks

e to the power of x squared end exponent cos e c space y fraction numerator d y over denominator d x end fraction equals x space sin space y semicolon 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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4a
Sme Calculator8 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

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4b
Sme Calculator3 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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5a
Sme Calculator2 marks

Consider the standard logistic equation

 fraction numerator d P over denominator d t end fraction equals k P open parentheses a minus P close parentheses 

where P is the size of a population at time t greater or equal than 0,  and where k and a are positive constants.  Let the population at time t equals 0 be denoted by P subscript 0. 

Write down the solution to the logistic equation in the case where P subscript 0 equals a, using mathematical reasoning to justify your answer.

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5b
Sme Calculator8 marks

In the case where P subscript 0 not equal to alpha, show that the solution to the logistic equation is

P open parentheses t close parentheses equals fraction numerator a A e to the power of a k t end exponent over denominator 1 plus A e to the power of a k t end exponent end fraction  

where A is an arbitrary constant.

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5c
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In the case where P subscript 0 not equal to alpha, write down an expression for A in terms of a and P subscript 0.

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5d
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In the case where P subscript 0 not equal to 0, determine the behaviour of P as t becomes large.

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5e
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In the case where 0 less than 2 P subscript 0 less than a, determine the value of t at which the initial population will have doubled.  Your answer should be given explicitly in terms of a comma k spaceand P subscript 0.

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6
Sme Calculator8 marks

Solve the differential equation

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

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7a
Sme Calculator9 marks

Consider the differential equation

x squared y apostrophe equals y squared plus 3 x y minus 8 x squared 

with the boundary condition y open parentheses 1 close parentheses equals negative 3. 

Solve the differential equation for y which satisfies the given boundary condition, giving your answer in the form y equals f open parentheses x close parentheses.

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7b
Sme Calculator3 marks

Determine the asymptotic behaviour of the graph of the solution as x becomes large.

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8
Sme Calculator7 marks

Solve the differential equation

open parentheses 4 x squared plus 1 close parentheses y apostrophe plus y equals fraction numerator 1 minus x plus 4 x squared minus 4 x cubed over denominator square root of e to the power of arctan space 2 x end exponent end root end fraction

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9a
Sme Calculator3 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 .

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9b
Sme Calculator7 marks

Solve the differential equation analytically, for y which satisfies the given boundary condition.

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9c
Sme Calculator3 marks
(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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10a
Sme Calculator3 marks

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

 fraction numerator d x over denominator d t end fraction equals fraction numerator sin space t over denominator 1 plus cos squared t end fraction comma space space space space space space space space space space space 0 less or equal than t less or equal than 3 

At time  t equals 1.6 comma space x equals 1. 

By using Euler’s method with a step length of 0.04, find an approximate value for x at time t equals 1.8.

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10b
Sme Calculator7 marks

The diagram below shows a graph of the exact solution  x equals f open parentheses t close parentheses to the differential equation with the given boundary condition.

 q10b_5-10_differential-equations_veryhard_ib_aa_hl_maths_diagram

Given that the graph of x equals f open parentheses t close parentheses  has exactly one point of inflection, find the exact value of the t-coordinate of the point of inflection.

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10c
Sme Calculator3 marks

Hence determine whether the approximation found in part (a) will be an overestimate or an underestimate for the true value of x when t equals 1.8.  Be sure to use mathematical reasoning to justify your answer.

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