Separable ODEs also arise in analysis of a fluid leaking from a tank.

This is regulated by Torricelli's Law which relates h,

the height of the liquid with A, the cross sectional area at the top.

The law states that the rate of change of h with respect to t is proportional

to the square root of h, and inversely proportional to A.

If we consider say a cone, or

anything where the cross sectional area is quadratic in h by some constant.

Maybe pi, maybe not.

Then we obtain h prime is proportional to h to the negative three halves.

This is separable.

And moving the h to three halves over to the left side.

And then integrating both sides, yields on the left,

h to the five-halves times two-fifths.

And on the right, a constant times t plus an integration constant.

Solving for h,

gives a linear function of t raised to the two-fifths

power where the constants will depend on the physics in your initial condition.

What this implies is that h approaches zero very rapidly at the end.

How rapidly?

It is in O of the final time minus t raised

to the two-fifths power, that comes into zero very very sharply,

when you're looking at something to a power of two-fifths.

Let's move on to a different class of nonautonomous ODEs.

These are the linear nonautonomous ODEs.

There of the form dx dt equals A of t times x

times B of t where A and B are functions of t.

For example, dx dt equals tx plus one half t squared is of this form.

We say that it's a linear equation because if you suppress the t

dependence you get something of the form dxdt equals ax plus b.

That right hand side should remind you of the equation.

The line, well, how do we solve such a problem?

The method that we will introduce is that of the integrating factor.

It's a little tricky so hang on.