Revision:Differential Equations

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A differential equation is an equation which contains a derivative in (such as dy/dx). When given a differential equation, you will often be asked to 'solve' the differential equation or find the 'general solution'. This basically means find an expression which does not contain any derivatives. To do this you will need to integrate.

Example

Here is a popular one that appears in exams quite frequently,

According to Newton's law of cooling, the rate at which the temperature of a body falls is proportional to the amount by which the temperature exceeds that of it's surroundings.

A room is at a constant temperature of . An object has temperature when it is brought into the room and 5 minutes later it's temperature is . What will it's temperature be after a further interval of 5 minutes?

Solution:

Let be the temperature of the object at time minutes after being brought into the room.

Let be the positive constant of proportionality.

We have in the formula because the temperature is falling.

Rearranging to put on the left hand side,

Integrating both sides with respect to ,

, by letting .

We know that when ,.

We use this to find A.

, so

When , ,so

So the differential equation becomes,

When ,

After a further 5 mintues the temperature of the object is .