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**Case 4**: *r*_{1} ≠ *r*_{2}, *k*_{1}>0

This will be a combination of cases 2 and 3. We now have the equation

I.e. an equation of the form

As you might see, if you try, this is not separable. It is a first-order linear differential equation, i.e. something that could fit into the pattern

In this case, we have that

Earlier on we have seen that this can be solved using an integrating factor,

This will be rather messy, so we will, for this time, start with an example, and come back to the general case a bit further on.

**Example**

Say we have Say we have *V*=*V*_{0}=2000 l, *r*_{1} = 15 l/s, *r*_{2} = 20 l/s, *k*_{1}=10/2000 kg/l and *y*_{0} = 50 kg. This would give us the equation

or

and thus

The integrating factor will now be

This will give us that

will become

or

This can now be rewritten as

Integrating both sides with respect to *t* will give us

or,

As before we can factorize out the 400 to get

To find *C*:

This will finally give us

This would be the second to lowest curve in the graph below.

**Back to the general case**

It will be easier to work with

than

We have

or

This gives us

and

or

and thus

that can be written as

Integrating this gives us

where *K*_{1} is the constant of integration.

This gives us

Filling in what we had initially we will get

or

Here we did the same trick as before, pulling out factors of *V*_{0}, and then letting them be absorbed by the constant of integration.

Finally, we find *K*:

This will give us:

We can see that case 2 is just a special case of case 4.

**When general is not general**

This is an example of when the “general” solution will not cover all cases. If one had started with the “general” case, one would end up with the solution above, but as one can see, this cannot be the general solution. Why not? Because we need to divide by *r*_{2}–*r*_{1}, and that would be 0 if *r*_{2}=*r*_{1}, we must thus examine that case too, and that would lead us to cases 1 and 2.

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