Differential Equations (DP IB Maths: AA HL)

Consider the first-order differential equation

Solve the equation given that  when ,  giving your answer in the form .

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

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

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 . 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.

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

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 of dragons in the islands may be modelled by the logistic equation

where  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.

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

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

Consider the differential equation

Explain why the substitution would be an appropriate method to use to solve the differential equation.

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

where  is an arbitrary constant.

Find the precise solution to the differential equation given that when .

Use the substitution  to solve the differential equation

for which satisfies the boundary condition . Give your answer in the form  .

Use an integrating factor to solve the differential equation

Consider the differential equation

with the boundary condition . 

Apply Euler’s method with a step size of  to approximate the solution to the differential equation at .

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

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

At time  , .

By using Euler’s method with a step length of 0.25, find an approximate value for  at time .

The diagram below shows a graph of the exact solution   to the differential equation with the given boundary condition.

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

Consider the first-order differential equation

Solve the equation given that  when  ,  giving your answer in the form  .       .

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

Solve each of the following differential equations for  which satisfies the given boundary condition, giving your answers in the form .

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 is proportional to the number, , of radioactive atoms currently remaining.  The amount of time, , 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 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  may be expressed as

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.

Consider the standard logistic equation

where  is the size of a population at time ,  and where  and  are positive constants.  Let the population at time  be denoted by . 

Write down the solution to the logistic equation in the case where , using mathematical reasoning to justify your answer.

In the case where , show that the solution to the logistic equation is

  

where  is an arbitrary constant.

In the case where , write down an expression for  in terms of  and .

In the case where , determine the behaviour of  as  becomes large.

In the case where , determine the value of  at which the initial population will have doubled.  Your answer should be given explicitly in terms of and .

Solve the differential equation

Consider the differential equation

with the boundary condition . 

Solve the differential equation for which satisfies the given boundary condition, giving your answer in the form .

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

Solve the differential equation

Consider the differential equation

with the boundary condition .
Apply Euler’s method with a step size of   to approximate the solution to the differential equation at  .

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

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

At time   

By using Euler’s method with a step length of 0.04, find an approximate value for  at time .

The diagram below shows a graph of the exact solution   to the differential equation with the given boundary condition.

Given that the graph of   has exactly one point of inflection, find the exact value of the -coordinate of the point of inflection.

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