Begin2.DVI

Figure 10-6. Four terminal networks In figure 10-6 the quantities I 1 , E 1 and I 2 , E 2 are the input and output current and v ...

Example 10-33. For the four terminal network illustrated one must have ( E 1 I 1 ) = ( T 11 T 12 T 21 T 22 )( E 2 I 2 ) =⇒ E 1 = ...

The investment of money at compound interest is an example of a physical problem which requires analysis of discrete values. Say ...

replaced by differences (discrete change), then linear ordinary differential equations become linear difference equations. Let u ...

Define the second forward difference as a difference of the first forward difference. A second difference is denoted by the nota ...

is an example of a first-order difference equation. The symbols L 1 (), L 2 () are operator symbols representing linear operator ...

Example 10-35. The function kN=k(k−1)(k−2) ···[k−(N−2)][k−(N−1)], k^0 ≡ 1 is called a factorial falling function which is a poly ...

Special Differences The table 10.1 contains a list of some well known forward differences which are useful in many applications. ...

Table 10.2 Some selected finite integrals 1. ∆−^1 ak= a k a− 1 a
= 1 2. ∆−^1 kn = kn+1 n+ 1 k n is factorial falling 3. ∆−^1 sin ...

Example 10-37. Evaluate the sum S= 1 ·2 + 2 ·3 + 3 ·4 + ···+n(n+ 1) Solution: Let fk=k(k+ 1) = k^2 +kand show one can write fkas ...

Solution: In the given difference equation, replace nby n+ 1 in all terms, to obtain yn+2 =yn+1 + 2yn, then one can verify n= 0, ...

Example 10-39. (Characteristic equation with real roots) Solve the second-order difference equation yk+2 − 3 yk+1 + 2 yk= 0. Sol ...

Example 10-40. (Characteristic equation with repeated roots) Find the general solution to the difference equation yn+2 − 4 yn+1 ...

are klinearly independent solutions of the difference equation. To show this, solve the difference equation (E−a)kyn= 0, (10 .58 ...

Figure 10-9. Polar form of complex numbers. In this figure ris called the modulus or length of the complex number λand θis calle ...

to obtain the real solutions y 3 (n) = rncos nθ y 4 (n) = rnsin nθ. The general solution is any linear combination of these func ...

member of the new set Salso occurs in the complementary solution, then members of the set Sare modified again. This is analogous ...

where αn, β nare to be determined. There are two unknowns and consequently two conditions are needed to determine these quantiti ...

where Cn+1 =un+1vn+2 −un+2vn+1 is called the Casoratian (the analog of the Wron- skian for continuous systems). It can be shown ...

Exercises In the following exercises if the size of the matrix is not specified, then assume that the given matrices are square ...