If we maintain the assumption that all the parameters are constant over time, this Great Lakes differential equation problem leads to a steady state problem which can be solved with MATLAB.
Since differential equations are difficult to solve researchers often content themselves with determining the long-run behavior of systems of differential equations. This is called steady state analysis. This analysis is easier to perform than the solution of the differential equations because steady state analysis involves solving an algebraic system of equations for specific numbers while the solution of differential equations involves finding formulas for functions, or at least function values at many different times.
The goal of steady state analysis is to find the values of each
unknown function (the 
's in our example) which cause the
system to enter a steady state where nothing further changes as time
progresses. The assumption is that the system being described will
approach such a steady state solution if enough time is allowed to
pass.
The key to finding the steady state solution of a system of
differential equations is to replace each occurence of the unknown
functions (
's in our case) with its (unknown) steady state
value. For our example we will call these steady state values
. To indicate that no change occurs once the system is in
the steady state we set all the derivatives to the value zero. The
result is a system of equations involving only variables, no unknown
functions or their derivatives. If this system is linear, as it will
be in our example, MATLAB can be used to solve for the 
's.
Considering the Great Lakes example, if we make the substitutions discussed above, we get the following algebraic system of equations:
The only unknowns in this system are the values of the various
's. Everything else is a known parameter in the problem.
Rearranging the terms of the equations to separate those which involve
an unknown and those which are constants,
we see this system in the
traditional form for a linear system, namely:
MATLAB's role in this endeavor is easier to see if we rewrite this linear system in a form where each variable is in its own column. For the Great Lakes problem this rearrangement yields:
In this last form it is easier to see that this linear system has the usual form for a linear system, Ax=b. The matrix A and vector b are also easily determined when the system is written in this form.
This linear system can be solved with MATLAB or, given its particularly convenient structure, can be solved by hand. If you are modeling some other system besides the simple Great Lakes system, say a system where several lakes drain into a central lake, the linear system will be more complicated and MATLAB (or some other linear system solver) will be required.