lecture_2 - MA 36600 LECTURE NOTES WEDNESDAY JANUARY 14...

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MA 36600 LECTURE NOTES: WEDNESDAY, JANUARY 14 Basic Mathematical Models (cont’d) Slope Fields. It is not necessary to find the exact solution of a differential equation in order to understand how the solution behaves. We discuss this concept through several examples. Consider a plot of x ( t ) versus t i.e. we plot the position x on the vertical axis and time t on the horizontal axis. Through any two points on the curve we can draw a line segment. Recall that given two distinct points ( t 0 , x 0 ) and ( t 1 , x 1 ), the line through them has slope slope of secant line = Δ x Δ t = x ( t 1 ) - x ( t 0 ) t 1 - t 0 . Of course, if the points are close to each other, we find the slope of the tangent line: slope of tangent line = lim t 1 t 0 x ( t 1 ) - x ( t 0 ) t 1 - t 0 = dx dt ( t 0 ) . In particular, the slope of the secant line is an approximation of the slope of the tangent line: Δ x Δ t dx dt = Δ x dx dt Δ t. Conversely, if we knew the slope x 0 ( t ) at every point ( t, x ), we could do a reasonable job of sketching the path x ( t ) vs. t. This is called the direction field or slope field . We can best illustrate this through some examples. Say that x 0 ( t ) = 0 for all time t . That means we have an object with no velocity, so we expect x ( t ) = x 0 to be a constant. In Figure 1, we find a plot of
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