TABLE OF CONTENTS A variety of important problems in science and engineering are nonlinear in nature. As discussed in section 2.1, except for some special cases they cannot be solved analytically. However, in many cases, a good deal of qualitative information about the solution can be extracted without actually solving the problem. Such information is of great value, because it not only provides insight into the problem, but also serves as a check on the numerical solution which is generally obtained using a computer.
As a motivation for this study, let us first consider two simple problems, one linear and the other nonlinear. It will become clear soon that nonlinear problems exhibit certain surprising features which are not found in linear ones.
The linear ODE
with IC
where a and b are constants, has the unique solution
If the problem arises physically, then a > 0, and let us restrict our discussion to this case. Then eq. (2.8.3) implies that as t ? ?,
where u s refers to the steady-state (also called equilibrium or critical) value of u. Now, u s could also be obtained directly from eq. (2.8.1), because in the steady state, by definition, u does not vary with t. Thus setting du/ dt in eq. (2.8.1) equal to zero gives u s = b/ a.
The two points to note about the above linear...
