Coupled differential equations

A scientist is studying the effect of introducing a population of type A bacteria into a population of type B bacteria.

At time t days, the number of type A bacteria, x, and the number of type B bacteria, y, are modelled by the differential equations

Show that

Determine a general solution for the number of type A bacteria at time t days.

Determine a general solution for the number of type B bacteria at time t days.

The model predicts that, at time T hours, the number of bacteria in the two populations will be equal.

Given thatx = 100 and y = 275 when t = 0

determine the value of T, giving your answer to 2 decimal places.

Suggest a limitation of the model.

A scientist is studying the effect of introducing a population of type A bacteria into a population of type B bacteria.

At time t days, the number of type A bacteria, x, and the number of type B bacteria, y, are modelled by the differential equations

Show that

Determine a general solution for the number of type A bacteria at time t days.

Determine a general solution for the number of type B bacteria at time t days.

The model predicts that, at time T hours, the number of bacteria in the two populations will be equal.

Given thatx = 100 and y = 275 when t = 0

determine the value of T, giving your answer to 2 decimal places.

Suggest a limitation of the model.

Two compounds, X and Y, are involved in a chemical reaction. The amounts in grams of these compounds, t minutes after the reaction starts, are x and y respectively and are modelled by the differential equations

Show that

Find, according to the model, a general solution for the amount in grams of compound X present at time t minutes.

Find, according to the model, a general solution for the amount in grams of compound Y present at time t minutes.

Given that x = 2 and y = 5 when t = 0

find

the particular solution for x,

the particular solution for y.

A scientist thinks that the chemical reaction will have stopped after 8 minutes.

Explain whether this is supported by the model.

Two compounds, X and Y, are involved in a chemical reaction. The amounts in grams of these compounds, t minutes after the reaction starts, are x and y respectively and are modelled by the differential equations

Show that

Find, according to the model, a general solution for the amount in grams of compound X present at time t minutes.

Find, according to the model, a general solution for the amount in grams of compound Y present at time t minutes.

Given that x = 2 and y = 5 when t = 0

find

the particular solution for x,

the particular solution for y.

A scientist thinks that the chemical reaction will have stopped after 8 minutes.

Explain whether this is supported by the model.

A scientist is studying the effect of introducing a population of white-clawed crayfish into a population of signal crayfish. At time t years, the number of white-clawed crayfish, w, and the number of signal crayfish, s, are modelled by the differential equations

=Show that

Find a general solution for the number of white-clawed crayfish at time t years.

=Find a general solution for the number of signal crayfish at time t years.

The model predicts that, at time T years, the population of white-clawed crayfish will have died out.

Given that w = 65 and s = 85 when t =

find the value of T, giving your answer to 3 decimal places.

=Suggest a limitation of the model.

A scientist is studying the effect of introducing a population of white-clawed crayfish into a population of signal crayfish. At time t years, the number of white-clawed crayfish, w, and the number of signal crayfish, s, are modelled by the differential equations

Show that

Find a general solution for the number of white-clawed crayfish at time t years.

Find a general solution for the number of signal crayfish at time t years.

The model predicts that, at time T years, the population of white-clawed crayfish will have died out.

Given that w 65 and s 85 when t

find the value of T, giving your answer to 3 decimal places.

Suggest a limitation of the model.