552 Ordinary Differential Equations
Exercise s 5.4 The Unit Function and Time-De lay Function
1. a. fi(x)=2u(x)-u(x- 1)
b. h(x) = u(x - 2) - u(x - 4)
c. h(x) = u(x - l)(x - 1)^2
d. f4(x) = u(x - l)(x^2 - 2x + 3)
e. fs(x)=u(x- 1f)sin3(x-1f)
f. f5(x) = u(x)x + u(x - l)(x - 1)
g. h(x) = [u(x) - u(x - l )]x
3. a. u(x - l)c^2 <x-l)
b. x - u(x - 2)(x - 2)c. u(x - 7r) cos 3(x - 7r)d. u(x - 7r) cosh 3(x - 7r)e. u(x - 2)e-(x-^2 J sin(x - 2)f. u(x - 3)e-<x-^3 ) [cos(x - 3) - sin(x - 3)g. u(x - 3)(ex-^3 - e-^3 (x-^3 l)/4
h. u(x - l)(x - l)ex-lExercises 5.5 Impulse Function
l. y = u(x - 2)e-^3 <x-^2 J- y = [u(x - 7r) sin 3(x - 7r) - u(x - 37f) sin 3(x - 31f)]/3
- y = [lOu(x - 1f)ex-11" sin2(x - 7r) + 4cosx - 2sinx + 16ex cos2x
-7exsin2x]/20 - y = [f(7r)u(x-1f)sinax]/a