The solutions obtained in this section can be used to check the results of a numerical solution technique. In addition, these solutions have also pointed out the following facts that will be helpful for properly using the numerical techniques presented in the next section.

  1. For certain types of differential equations, called linear equations, the characteristic polynomial can be found by making the substitution y(t) = Ae’.
  2. If any of the characteristic roots has a positive real part, the equation is unstable. If all the roots have negative real parts, the equation is stable.
  3. If the equation is stable, the time constants can be found from the negative reciprocal of the teal parts of the characteristic roots.
  4. The equation’s largest time constant indicates how long the solution takes to reach steady state.
  5.  The equation’s smallest time constant indicates how fast the solution changes with t.
  6. The frequency of oscillation of the free response can be found from the imaginary parts of the characteristic roots.
  7. The rate of change of the forcing function affects the rate of change of the solution. In particular, if the forcing function oscillates, the solution of a linear equation will also oscillate and at the same frequency.
  8.  The number of initial conditions needed to obtain the solution equals the order of the equation.
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