**SUMMARY**

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.

- For certain types of differential equations, called linear equations, the characteristic polynomial can be found by making the substitution y(t) = Ae’.
- 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.
- If the equation is stable, the time constants can be found from the negative reciprocal of the teal parts of the characteristic roots.
- The equation’s largest time constant indicates how long the solution takes to reach steady state.
- The equation’s smallest time constant indicates how fast the solution changes with t.
- The frequency of oscillation of the free response can be found from the imaginary parts of the characteristic roots.
- 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.
- The number of initial conditions needed to obtain the solution equals the order of the equation.

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