Differential equations and modelling

Find the general solution of the differential equation

dydx = (2x + 1)y,   y > 0

giving your answer in the form y = f(x).

Given that y = 2 when x = 0, solve the differential equation

dydx = e^2xy,   y > 0

giving your answer in the form y = f(x).

A scientist grows bacteria in a dish. The number of bacteria, N, t hours after the start of the experiment increases at a rate proportional to N. Initially there are 500 bacteria and after 3 hours there are 2000 bacteria.

1. By forming and solving a differential equation, show that N = 500e^kt, where k is a positive constant.
2. Find the exact value of k.
3. Find, according to the model, the time taken for the number of bacteria to reach 10 000. Give your answer in hours to 3 significant figures.

The variables x and y satisfy the differential equation

dydx = y tan x,   0 ≤ x < π2,   y > 0

Given that y = 3 when x = 0,

1. show that y = 3 sec x
2. find the exact value of x for which y = 6

The variables x and y satisfy the differential equation

dydx = x e^−y

and y = 0 when x = 2

1. Solve the differential equation, giving your answer in the form y = ln f(x)
2. State the set of values of x, with x > 0, for which your solution is defined.

A young tree is planted in a park. Its height, h metres, t years after planting is modelled by the differential equation

dhdt = 6 − ht + 1,   t ≥ 0,   1 ≤ h < 6

The tree is 1 metre tall when it is planted.

1. Show that h = 6 − 5t + 1
2. Find, according to the model, the height of the tree 4 years after planting.
3. State the height that the model predicts the tree approaches in the long term.

A block of ice is left to melt. Its mass, m kg, t hours after it was put out is modelled as decreasing at a rate proportional to the square root of m. The initial mass is 100 kg and after 5 hours the mass is 64 kg.

1. Write down a differential equation linking m and t.
2. Solve your differential equation to show that m = (10 − 0.4t)²
3. Find, according to the model, the time taken for the ice to melt completely.

A mug of tea is made in a kitchen where the room temperature is a constant 20 °C. The temperature of the tea, θ °C, t minutes after it is made is modelled by

dθdt = −k(θ − 20)

where k is a positive constant. The tea is made at 90 °C and after 10 minutes its temperature is 60 °C.

1. Show that θ = 20 + 70e^−kt
2. Show that k = 110 ln 74
3. Find, according to the model, the time taken for the tea to cool to 30 °C. Give your answer to the nearest minute.
4. State what happens to the temperature of the tea in the long term, according to the model.

Water is pumped into a garden pond at a constant rate of 0.6 m³ per hour. At the same time water leaves the pond through an overflow pipe at a rate of 0.2V m³ per hour, where V m³ is the volume of water in the pond t hours after the pump is switched on.

1. Show that 5dVdt = 3 − V
2. Given that V = 1 when t = 0, show that V = 3 − 2e^−0.2t
3. Find, according to the model, the time taken for the volume to reach 2.5 m³. Give your answer in hours to 3 significant figures.
4. Explain why, according to the model, the volume of water in the pond can never reach 3 m³.

After a storm a puddle on a flat playground is modelled as a circle. At time t hours the radius of the puddle is r cm and its area is A cm². In the model the area decreases at a rate inversely proportional to r.

1. Show that drdt = −cr², where c is a positive constant.
2. Given that the radius is 30 cm at t = 0 and 20 cm at t = 1, show that r³ = 27 000 − 19 000t
3. Find, according to the model, the time taken for the puddle to dry up completely. Give your answer to the nearest minute.

The number of fish, P hundred, in a lake t years after a group of fish was released into it is modelled by the differential equation

dPdt = P(10 − P)20,   0 < P < 10

The group released contained 200 fish.

1. Express 1P(10 − P) in partial fractions.
2. Show that P = 101 + 4e^−0.5t
3. Find, according to the model, the time taken for the number of fish to reach 800. Give your answer in the form a ln 2 years, where a is an integer.
4. State the maximum number of fish the lake can support, according to the model.

A gardener measures a seedling. Its height, h cm, t weeks after the first measurement is modelled by the differential equation

dhdt = t e^−0.5t √h,   t ≥ 0

When t = 0 the seedling is 4 cm tall.

1. Show that √h = 4 − (t + 2)e^−0.5t
2. Find, according to the model, the height of the seedling 4 weeks after the first measurement. Give your answer to 3 significant figures.
3. Find the height that the model predicts the seedling approaches in the long term, and give one reason why the model may not be realistic for a real plant.