Find the general solution to the differential equation
where .
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Find the general solution to the differential equation
where .
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Find the general solution to the differential equation
where .
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The differential equation
is used to model the rate at which water is leaking from a container, where
is the volume of water in the container
is the time in seconds
is a positive constant
Explain, in context, the significance of the negative sign in the model.
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Find the general solution to the differential equation.
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Given that
the initial volume of the container is 300 litres
find a complete equation linking and
.
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Given that , find the general solution to the differential equation
writing your answer in the form
where is a constant and
is a function of
which you should find.
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Given that , find the general solution to the differential equation
writing your answer in the form
where is a constant and
is a function of
which you should find.
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The volume of water in a sink, , decreases with time
, measured from the point at which the plug is removed.
It is known that decreases at a rate proportional to its volume.
Use this information to write down a suitable differential equation for and
, using a constant of proportionality
where
.
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The general solution to the differential equation in part (a) can be written in the form
where .
(i) State, in the context of the question, what the constant represents.
(ii) Briefly explain the significance of the negative sign in the solution.
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A differential equation is given by
where when
.
Show that
where is a constant to be found.
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A differential equation is given by
where when
.
Show that
where is a constant to be found.
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A large spherical balloon is deflating.
At time seconds the balloon has radius
cm and volume
cm3.
The volume of the balloon is modelled as decreasing at a constant rate.
Using this model, show that
where is a positive constant.
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Given that
the initial radius of the balloon is 40 cm
after 5 seconds the radius of the balloon is 20 cm
the volume of the balloon continues to decrease at a constant rate until the balloon is empty
solve a differential equation to find a complete equation linking and
.
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Find the limitation on the values of for which the equation in part (b) is valid.
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By solving the differential equation, show that the general solution to
where is
where and
are constants.
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Find the general solution to the differential equation
giving your answer in the form where
is a constant and
is a function to be found.
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Find the general solution to the differential equation
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Find the particular solution to the differential equation
given that the graph of against
passes through the point with coordinates
.
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A differential equation is given by
It is known that when
.
Solve the differential equation, giving your answer in the form
where and
are rational numbers to be found.
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A differential equation is given by
where when
.
Solve the differential equation, giving your answer in the form
where is a function to be found.
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A weather balloon of volume m3 is being inflated, where
is the time in minutes after inflation begins.
The rate of change of its volume is inversely proportional to its volume
When the rate of inflation of the balloon is 10 m3 min-1, the volume of the balloon is 20 m3
Use this information to write down a suitable differential equation for and
.
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Show that the general solution to the differential equation is
where is a constant.
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Initially, the balloon is flat with a volume of 0 m3.
Find the volume of the balloon after 25 minutes.
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A disease affecting trees is spreading throughout a large forested area. Let be the number of infected trees
days after the disease was first discovered.
A model for and
is given by
where is a positive constant.
It is known that
When the disease was first discovered, 3trees were infected
Ten days after the disease was first discovered, 10 trees were infected
Solve the differential equation to show that
where
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Scientists believe the majority of the forest can be saved from infection if action is taken before 30 trees are infected.
Find the number of days (since first discovering the disease) that the model predicts scientists have in order to take action.
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A bar of soap in the shape of a cuboid is placed in a bowl of warm water and its volume is recorded at regular intervals. The water is maintained at a constant temperature.
Before being placed in the water, the soap measures 3 cm by 6 cm by 10 cm
Two minutes later, the bar of soap measures 2.85 cm by 5.7 cm by 9.5 cm
The rate of decrease in volume of the bar of soap is modelled as being directly proportional to its volume
Defining any variables you use, find and solve a differential equation linking the volume of the bar of soap and time.
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By considering the volume of soap as time increases, suggest a limitation of the model.
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Express in partial fractions.
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When chemical and chemical
are mixed, oxygen is produced.
A scientist mixed these two chemicals and measured the total volume of oxygen produced over a period of time.
The total volume of oxygen produced, m3,
hours after the chemicals were mixed, is modelled by the differential equation
where is a constant.
Given that exactly 2 hours after the chemicals were mixed, a total volume of 3 m3 of oxygen had been produced, solve the differential equation to show that
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The scientist noticed that
there was a time delay between the chemicals being mixed and oxygen being produced
there was a limit to the total volume of oxygen produced
Deduce from the model
(i) the time delay giving your answer in minutes,
(ii) the limit giving your answer in m3
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A tank in the shape of a cuboid is being filled with water.
The base of the tank measures 20 m by 10 m and the height of the tank is 5 m, as shown in Figure 1.
At time minutes after water started flowing into the tank the height of the water was
m and the volume of the water in the tank was
m3.
In a model of this situation
the sides of the tank have negligible thickness
the rate of change of is inversely proportional to the square root of
Show that
where is a constant.
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Given that
initially the height of the water in the tank was 1.44 m
exactly 8 minutes after water started flowing into the tank the height of the water was 3.24 m
use the model to find an equation linking with
, giving your answer in the form
where and
are constants to be found.
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Hence find the time taken, from when water started flowing into the tank, for the tank to be completely full.
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Find the general solution to the differential equation
where , giving your answer in the form
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Find the general solution to the differential equation
giving your answer in the form .
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Show that the general solution to the differential equation
is given by
where is a constant.
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Solve the differential equation
given that when
.
Give your answer in the form
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A hot air balloon is being inflated at a rate that is inversely proportional to the square of its volume.
Defining variables for the volume of the balloon (m3) and time (seconds), write down a differential equation to describe the relationship between volume and time as the hot air balloon is inflated.
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You are given the following information:
Initially, the hot air balloon has a volume of zero
After 400 seconds of inflating, its volume is 600 m3
The hot air balloon is considered ready for release when its volume reaches 1250 m3
If the hot air balloon needs to be ready for release by midday, find the latest time that it can start being inflated.
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Find the general solution to the differential equation
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Find the particular solution to the differential equation
where the graph of against
passes through the point with coordinates
.
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A petrol tanker is leaking petrol. The volume of petrol inside the petrol tanker is litres, which is decreasing over time
minutes.
The volume of petrol inside the tanker decreases at a rate proportional to the square of its volume
The initial volume of petrol inside the tanker is 4000 litres
After 10 minutes, the volume of petrol in the tanker has dropped by 30%
By forming and solving a suitable differential equation, show that
where and
are constants to be found.
Hence, describe what happens to the volume of petrol after a long time.
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A spherical mint of radius 5 mm is placed in the mouth and sucked.
Four minutes later, the radius of the mint is 3 mm.
In a simple model, the rate of decrease of the radius of the mint is inversely proportional to the square of the radius.
Using this model and all the information given, find an equation linking the radius of the mint and the time.
(You should define the variables that you use.)
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Hence find the total time taken for the mint to completely dissolve. Give your answer in minutes and seconds to the nearest second.
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Suggest a limitation of the model.
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Use the substitution to show that
where is a constant.
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A team of scientists is studying a species of slow growing tree.
The rate of change in height of a tree in this species is modelled by the differential equation
where is the height in metres and
is the time, measured in years, after the tree is planted.
Find, according to the model, the range in heights of trees in this species.
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One of these trees is one metre high when it is first planted.
According to the model, calculate the time this tree would take to reach a height of 12 metres, giving your answer to 3 significant figures.
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Water flows at a constant rate into a large tank.
The tank is a cuboid, with all sides of negligible thickness.
The base of the tank measures 8 m by 3 m and the height of the tank is 5 m.
There is a tap at a point at the bottom of the tank, as shown in Figure 5.
At time minutes after the tap has been opened
the depth of the water in the tank is metres
water is flowing into the tank at a constant rate of m3 per minute
water is modelled as leaving the tank through the tap at a rate of m3 per minute
Show that, according to the model,
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Given that when the tap was opened, the depth of the water in the tank was 2 m, show that, according to the model,
where ,
and
are constants to be found.
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Given that the tap remains open, determine, according to the model, whether the tank will ever become full, giving a reason for your answer.
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Find the general solution to the differential equation
where , giving your answer in the form
.
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Palm trees are being planted on an island. Let be the total number of palm trees planted on the island after
days.
The variables and
are modelled by the differential equation
where and
is a positive constant.
By solving the differential equation, show that
where is a positive constant.
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It is known that
Initially 2 palm trees are planted
After 14 days, 4 palm trees in total have been planted
Use this information to show that
where is a rational number to be found.
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By considering the form of the solution to the differential equation, suggest a range of values of for which the model is valid.
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The temperature of a heated object, °C, cools over time,
minutes. The room temperature (called the ambient temperature) is constant,
, where
.
Newton’s Law of Cooling states that the rate of decrease in temperature of a heated object is directly proportional to the difference between the object’s temperature and the ambient temperature.
By forming and solving a differential equation in and
(involving the constant
and a positive constant of proportionality,
) show that
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For food safety reasons, a meat processing factory must store its products at a temperature of below -1 °C.
One particular product has a temperature of 7 °C
It is placed in one of the factory's freezers, which has a constant ambient temperature of -4 °C
One minute later, its temperature has dropped to 4.7 °C.
Any products that fail to cool to below -1 °C within 6 minutes must be discarded
Determine whether or not this product will need to be discarded.
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Show that the solution to the differential equation
where when
may be written in the form
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(i) Prove that if then
where is an integer.
(ii) Hence deduce that the particular solution to the differential equation in part (a) is
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