Method of Solution

The two equations we arrived at above are both ordinary differential equations. In principle, we should have enough tools at our disposal to solve them by hand, given enough time to eliminate all the errors. Before we decide how to attack them mathematically, however, let's rearrange them a bit.

For the circular cylinder, we can easily separate the variables, putting all functions of y on one side and all the constants and t on the other side:

Rearranging the equation for the conical tank requires a bit more algebra:

As noted above, r is the minor radius of the cone section, is the major radius, and is the radius of the drain.

Both of these equations appear to be soluble analytically, although (13) may be quite tedious. In either case, it may not be possible to find an explicit expression for y(t). That is, the solution may be in the form of a transcendental or some other function which does not allow us to isolate the solution on the left and the independent variables and constants on the right. In that case, it may make sense to evaluate t(y), if that is more easily plotted.