Example E Consider the first-order equation Yk+1 - ayk = Rr, (4.294) where a is a constant. Taking the z-transform of each term in this equation gives zF(2) – zyo – aF(2) = G(2), (4.295) where G(z) = Z(R). Solving for F(2), we obtain G(2) + zyo F(2) = (4.296) Z - a Suppose R = 0; then G(2) 0 and equation (4.296) becomes yỗ F(2) = (4.297) a From Table 4.1, we see that the inverse transform of z/(z-a) is ak. Therefore, the general solution to the difference equation Yk+1 - ayk = 0 (4.298) is Yk = Yoak. (4.299) as can be readily checked by using some other method of solution. Note that there is one arbitrary constant, namely, yo. LINEAR DIFFERENCE EQUATIONS 155 Now suppose R = 1, with a + 1. Therefore, from equation (4.296), we have yoz2 + (1 - yo)z (n - 2)( 2/(z - 1) + zYo F(2) = (4.300) z - a where G(2) = Z(1) (4.301) 2 -1 Therefore, writing F(2) 2(A+ B) - (aA + B) (z – 1)(z - a) В (4.302) 2 - a and comparing with equation (4.300) gives the following two equations for A and B: A +B = yo, (4.303) aA + B = yo - 1. Solving gives (1- a)yo - 1 B = A = (4.304) 1- a 1- a and 1 (1 — а)уo — 1 F(Z) (4.305) a z - 1 1- a From Table 4.1, we determine the inverse transform of equation (4.305) to be 1 (1-a)yo -1ak. (4.306) 1- a 1- a This is the general solution to the difference equation
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
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