5.2 Differential Equations

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.

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.

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).

Find the general solution to the differential equation

Find the general solution to the differential equation

giving your answer in the form .

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

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 m^3.  If the balloon needs to be ready for a midday release, what is the latest time that it can start being inflated?

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.

By separating the variables, show that the general solution to the differential equation

can be written as

where c is the constant of integration.

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.

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.

Given that , find the general solution to the differential equation

Find the general solution to the differential equation

giving your answer in the form .

Find particular solutions to the following differential equations, using the given boundary conditions.

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).

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?

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.

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.

A large container of water is leaking at a rate directly proportional to the square of the volume of water in the container.

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, T_amb °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.

Find the general solution to the differential equation

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

Show that the relationship between x and y in  may also be expressed by the set of all equations of the forms

where n is an integer.

Hence deduce that the particular solution to the differential equation in part (a), with the given boundary condition, is  .

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.