Find the general solution to the differential equation
Find the particular solution to the differential equation
given that the graph of S against x passes through the point (0 , 5).
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Find the general solution to the differential equation
Find the particular solution to the differential equation
given that the graph of S against x passes through the point (0 , 5).
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By separating the variables, show that the solution to the differential equation
can be found by solving
Show that the general solution to the differential equation in part (a) is
where c is a constant.
By letting , show that the general solution to the differential equation in part (a) can be written in the form
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The velocity of a particle is given by the differential equation
where s is the displacement of the particle in metres from a fixed point O at time t seconds. At time the particle is located at point O.
Write down the initial velocity of the particle.
Use the method of separating variables to find an expression for the displacement of the particle from O after t seconds.
Find and hence explain why the acceleration of the particle is constant.
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The differential equation
is used to model the rate at which water is leaking from a container.
V is the volume of water in the container at time t seconds.
k is a constant.
Explain the use of a negative sign on the right-hand side of the differential equation, and the impact this has on the value of the constant k.
Show that
and hence find the general solution to the differential equation.
Given that and the initial volume of the container is 300 litres, find the particular solution to the differential equation, giving your answer in the form
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Given that , find the general solution to the differential equation
giving your answer in the form
Given that , find the general solution to the differential equation
giving your answer in the form
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Use the given boundary condition to find the particular solution to the following differential equations:
, ,
, ,
giving your answer in the form .
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A large weather balloon is being inflated.
The rate of change of its volume, , where V m3 is the volume of the balloon t minutes after inflation commenced, is inversely proportional to its volume.
Show that the general solution to the differential equation found in part (a) is
where c is a constant.
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A tree disease is spreading throughout a large forested area.
The differential equation
where k is a positive constant, is used to model the number of infected trees, N, at a time t days after the disease was first discovered.
Show that
Hence show that
where c is a constant.
Given that and that when the disease was first discovered, four trees were infected.
Find the particular solution of the differential equation and use it to estimate the number of infected trees after 30 days.
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Find the general solution to the differential equation
Find the particular solution to the differential equation
given that the graph of V against x passes through the point (0, 3).
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By separating the variables, show that the general solution to the differential equation
can be written as
where c is the constant of integration.
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The velocity of a particle is given by the differential equation
where s is the displacement of the particle in metres from a fixed point O, and t is the time in seconds. At time the particle is located at point O.
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A large container of water is leaking at a rate directly proportional to the volume of water in the container.
Using the variables V, for the volume of water in the container, and t, for time, write down a differential equation involving the term , for the volume of water in the container.
The general solution to the differential equation in part (a) can be written in the form
where k is a positive constant.
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Given that , find the general solution to the differential equation
Find the general solution to the differential equation
giving your answer in the form .
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Find particular solutions to the following differential equations, using the given boundary conditions.
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A large weather balloon is being inflated at a rate that is inversely proportional to its volume.
Find the general solution to the differential equation found in part (a).
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A tree disease is spreading throughout a large forested area.
When the disease was first discovered, three trees were infected.
Ten days later ten trees were infected.
The differential equation
where k is a positive constant, is used to model the number of infected trees, N, at a time t days after the disease was first discovered.
Find the particular solution to the differential equation.
Scientists believe the majority of the forest can be saved from infection if action is taken before 30 trees are infected.
Measured from the time when the disease was first discovered, how many days does the model predict the scientists have to take action in order to save the majority of the forest from infection?
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Find the general solution to the differential equation
Find the particular solution to the differential equation
where the graph of V against x passes through the point (0, -4).
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Show that the general solution to the differential equation
is
where A is a constant.
On the same set of axes sketch the graphs of the solution for the instances where
In each case be sure to state where the graph intercepts the y-axis.
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The acceleration of a particle is given by the differential equation
where s is the displacement of the particle in metres from a fixed reference point O, and t is the time in seconds.
The particle starts its journey at point O at and has an initial velocity of 1 m s-1.
Find an expression for the displacement of the particle in terms of t.
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A large container of water is leaking at a rate directly proportional to the volume of water in the container.
Defining any variables, write down a differential equation that describes how the volume of water in the container varies with time.
By separating the variables, find the general solution to your differential equation from part (a).
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Find the general solution to the differential equation
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
can be written in the form
where A is a constant.
Find the particular solution to the following differential equation, using the given boundary condition
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A large weather 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 weather balloon is inflated.
Given that initially the balloon may be considered to have a volume of zero, and that after 400 seconds of inflating its volume is 600 m3, find the particular solution to your differential equation.
Although it can be inflated further, the balloon is considered ready for release when its volume reaches 1250 m3. If the balloon needs to be ready for a midday release, what is the latest time that it can start being inflated?
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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.
What happens to the volume of the bar of soap for large values of t?
Briefly explain why this could be considered a criticism of the model.
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Find the general solution to the differential equation
Find the particular solution to the differential equation
where the graph of V against x passes through the point (0,2).
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Show that the general solution to the differential equation
is
where A and k are constants.
On separate diagrams sketch a graph of the solution for in the instances when
On both diagrams state where the graph intercepts the y-axis.
You may assume in both cases.
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The acceleration of a particle moving in a straight line is given by the differential equation
where s m is the displacement of the particle relative to a fixed point O, and t is the elapsed time in seconds. The particle starts its journey with a velocity of -6 m s-1, from a point 50 m in the positive direction from the point O.
Find an expression for the displacement, s, of the particle in terms of t.
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A large container of water is leaking at a rate directly proportional to the square of the volume of water in the container.
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Newton’s Law of Cooling states that the rate of cooling of an object is directly proportional to the difference between the object’s temperature and the ambient temperature (temperature of the object’s surroundings).
By setting up and solving an appropriate differential equation, show that
where T °C is the temperature of the object, Tamb °C is the ambient temperature, t is time, and and A are both constants. You may assume in working out your solution that the ambient temperature is constant, and that the temperature of the object is greater than the ambient temperature.
A meat processing factory must store its products at a temperature below -1 °C.
Due to the production process, products, before cooling, typically have a temperature between 5 °C and 10 °C.
The company therefore has a policy that any products failing to cool to below -1 °C within 6 minutes of being processed must be discarded.
The factory stores its products in a freezer with a constant ambient temperature of -4 °C.
A product that has just finished being processed has a temperature of 7 °C and is immediately placed in the freezer. One minute later its temperature has dropped to 4.7 °C. Determine whether or not this product will need to be discarded.
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Find the general solution to the differential equation
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Using the standard integral result
(where k is a constant, and c is a constant of integration), show that the solution to the differential equation
with boundary condition, , may be written in the form
where n is an integer.
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A tree disease is spreading throughout a large forested area.
The rate of increase in the number of infected trees is modelled by the differential equation
where N is the number of infected trees, t is the time in days since the disease was first identified and k is a positive constant.
Solve the differential equation above, and show that the general solution can be written in the form
where A is a positive constant.
Initially two trees were identified as diseased.
A fortnight later, 4 trees were infected.
Using this information, find the values of the constants A and k.
By considering the solution to the differential equation along with the values of A and k found in part (b), suggest a range of values of t for which the model might be considered reliable.
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