Page 254 - Schaum's Outline of Differential Equations

CHAP. 23] CONVOLUTIONS AND THE UNIT STEP FUNCTION 237

23.9. Graph the function/(*) = u(x-2)-u(x-3).
Note that

Thus,

the graph of which is given in Fig. 23-4.

23.10. Graph the function/(*) = 5- 5u(x - 8) for x > 0.
Note that

Thus

The graph of this function when x > 0 is given in Fig. 23-5.

Fig. 23-4 Fig. 23-5

23.11. Use the unit step function to give an analytic representation of the function/(X) graphed in Fig. 23-6.
Note that f(x) is the function g (x) = x, x > 0, translated four units in the positive x-direction. Thus,
f(x) = u(x - 4)g (x-4) = (x- 4)u(x - 4).

23.12. Use the unit step function to give an analytic description of the function g(x) graphed on the interval
(0, co) in Fig. 23-7. If on the subinterval (0, a) the graph is identical to Fig. 23-2.
Let/(jc) represent the function graphed in Fig. 23-2. Then g(x) =f(x)[l - u(x - a)}.