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Understanding Simple Differential Equations

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A differential equation is any equation containing a derivative term. For example
or

More complicated examples of differential equations, that you will not meet until Further Maths are in the form
A First Order Differential Equation 

and
A Second Order Differential Equation

Separation Of Variables

The type of differential equations that you will face in A-level Maths, can be solved using a method called separating the variables. This involves collecting all y terms on one side of the equation, and all the x terms on the other side. You then integrate each side separately.

For Example

Solve the equation given that y=6 when x=4
To separate the variables, we need to multiply by dx. Since this will leave the dy and dx terms on their own, we must at this point include integral signs
We can now integrate each side separately. Remember that 
We therefore get You must remember to include a constant of integration.

To find the constant, you just need to substitute in the values that were given in the question.

The solution to the differential equation is therefore

Clearly this was a very simple example, but it shows the basic principles for solving simple differential equations. Over the next few pages more complicated examples will be shown.


 

One of the most common situations to give a differential equation, is that of growth and decay.

If the rate at which a population x grows is proportional to its size (x) then this can be described as
where k is the constant of proportionality.

This time, in order to collect terms on the same side, we not only have to multiply by dt, but we must also divide by x. This gives,
Integrating this gives
If we exponentiate each side

 

 

 We can separate the e terms

ec is just a constant.

This constant is just the value of x when t=0.

So, by forming a differential equation, you can derive the formula for exponential growth and decay(if k is negative).

 

A differential equation will obviously be more complicated to solve when it contains both x and y terms, and the integrations may not always be simple. The next two pages give examples of more difficult equations.


 
Solve the equation
given that y=0 when x=0

First we need to re-arrange the equation, to get similar terms on the same side.
To integrate the LHS, we will need to split into partial fractions.

 
Factorise the denominator.

Split into partial fractions

Multiply to write as one fraction

Collect together terms.

 
We can now equate the numerators
which gives 2 simultaneous equations
[1]
[2]
[2] can be rewritten
which substituted into [1] gives

 

 
We can now integrate
by writing it as
This implies

 

 

 

 You must not forget the minus sign in front of the first log.

Collect the logs together.

Tidy up.

 

We now just need to use the condition given in the question to find the constant c.
y=0 when x=0

 

 

Therefore our equation is

Solve the differential equation
given that
when

 
First separate the variables
which can be written

 
Integrating gives
Substitute in the given values

 

 

 You should know these trig ratios.

 

Tidy up to find c
Therefore the solution is

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