Prerequisite Knowledge:
Posted below are links to my lecture notes from Ma 241. We will review much of these materials and more. These notes are loosely based on the calculus text written by Stewart. In Ma 341 we will learn how to solve many additional examples, but the basic themes remain the same.

  • Introduction to differential equations (7.1)
  • [166] Differential equations are described in general. The definition of the terms: ordinary, order, autonomous, linear, homogeneous, nonhomogeneous and solution, are given and discussed as they apply to differential equations.
  • Direction Fields and Euler's Method (7.2)
  • [167-170] Basic terminology and graphing DEqs. Euler's method is then discussed. We explicitly see how to construct an approximate solution by using Euler's method. In essence, this is nothing more than tracing out a solution curve in the direction field. Equilibrium solutions are also defined and discussed in several examples.
  • Separation of variables (7.3)
  • [171-177] Separate then integrate. A number of physically interesting examples given.
  • Exponential growth (7.4)
  • [178] Perhaps the most naive growth model, yet it is a good approximation of many real world processes. We study the differential equation that it arises from and derive the solutions by math we've learned in earlier sections.
  • Logistic growth (7.5)
  • [179-182] The logistic equation is slightly less naive than simple exponential growth. We analyze the logistic differential eqn. in two ways. First, we study implications of the DEqn directly and find some rather interesting general conclusions about any solution. Second, we solve the logistic DEqn directly and find the general form of the solution. An example of how you might try to apply it is then given (realistic modeling of population growth has not proven to be very reliable historically, so I'll abstain from anything but the math here... )
  • Homogeneous 2nd order linear ordinary DEqns (7.7)
  • [183-187] We begin by carefully analyzing the possible solutions to the homogeneous case. We find three possiblities corresponding to the three types of solutions to the quadratic characteristic equation. In each of the cases I, II, and III we find two linearly independent fundamental solutions. The general solution is then formed by taking a linear combination of the fundamental solutions.
  • Nonhomogeneous 2nd order linear ordinary DEqns (7.8)
  • [183-187] We begin by reviewing the possible solutions to the homogeneous case. Next, we explain the form of the general solution to a nonhomogenous DEqn is the sum of a complementary and particular solution. The complementary solution is found by the same technique as in section 7.7. Then, we see how to find the particular solution through the method of undetermined coefficients. We begin with several examples, next a general algorithm describing the method is given (hopefully exposing some of the subtleties avoided in the first few examples), and after that yet more examples are given. Finally, we conclude by explaining how the complementary and particular solutions combine to make the general solution (a proof long overdue at this point in the notes).
  • Springs and RLC circuits (7.9)
  • [192-195] We study the motion springs in a viscous media and three cases result (under/over/critical damping), just like in the last section. It is the same math. Then we study springs that are pushed by an outside force and we encounter the interesting phenomenon of "resonance". Finally, we note the analogy between the RLC circuit and a spring with friction.

    On the basis of these notes I gave the following tests in MA241,
  • Test three solution
  • Test three solution

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