There are several situations where using Laplace transforms is easier than solving an equation using the techniques of chapter 2. One such situation is when you are more interested in certain qualitative features of the solution rather than the explicit formula for the solution. Some qualitative features of the solution can be read directly from the Laplace transform of the function. This means you will just have to carry out steps 1 and 2 of our paradigm to find the Laplace transform of the solution. Since taking the inverse Laplace transform in step 3 is the most complicated part of the paradigm, if we can skip this step, we can usually save some time and effort in answering our question.

The particular features that are easiest to read off the Laplace transform are whether the function tends to 0 or infinity as time tends to infinity. This is referred to as the stability of the solution. In addition, we can quickly tell whether the solution function is oscillating or not. We can also get a sense of how quickly the solution tends to 0 (if it does tend to 0).

The applet at the top of the page has two graph windows. The window on the right shows the graph of the solution curve to the equation defined at the top of the applet. The window on the left shows the poles of the Laplace transform of the solution function. The Laplace transform is a rational function, that is a quotient of two polynomials. The poles (as you may remember from algebra) are the zeros of the polynomial in the denominator of the Laplace transform of the function. The poles are marked with an X on the complex plane. If you get a double pole (a double root of the polynomial in the denominator), then the X will be circled.The differential equation is initially set to x'' + 9x = 0, x(0)=4, x'(0)=0. You can check that the Laplace transform of the solution is 4s/(s2+9) and so the poles are the roots of s2 + 9 = 0, which are s = ±3i.

Questions

1. The initial equation, x'' + 9x = 0, is undamped. Increase the coefficient of x' from 0 to 10 in steps of 2. How do the poles of the Laplace transform change as you increase the damping? How does the speed with which the solution converges to 0 change as you increase the damping?

2. Now experiment with changing the values of the different coefficients, while leaving the forcing function set at 0. How can you distinguish undamped, underdamped, critically damped, and overdamped equations from the poles of the Laplace transform of the solution?

3. Continue to experiment with changing the values of the coefficients of x'', x' and x, only now pay attention to the speed with which the solution tends to 0. How can you judge how quickly the solution tends to 0 from the poles of the Laplace transform?

4. In general, changing the values of the initial conditions won't change the poles in the left window. Can you explain why the initial values usually don't affect the locations of the poles of the Laplace transform? Can you find a second-order example (i.e. where the coefficient of x'' is not 0) where changing the initial values will cause just one of the poles to disappear? This question may be easier if you start with a Laplace transform with just one pole and work backwards to find the second-order initial value problem

5. Now change the amplitude of the forcing function (the right-hand-side of the equation) to something positive. This will add two more poles to the left window. So far we have only considered equations where the solution converges to 0, or at least stays bounded (in other words, we have only considered stable equations). Our solution will tend to infinity (and hence be unstable) if we have resonance. How can you distinguish resonance from the poles of the Laplace transform?

6. One other way to have solutions tending to infinity is to have a negative amount of friction. This may seem ridiculous, but consider the feedback loop when a microphone is placed in front of an amplifier. In this applet, it is possible to enter a negative value for the coefficient of x'. Where are the poles of the Laplace transform when the damping is negative? You will want to use a small negative damping to keep all the poles on the screen.