## Yr y j

and W4)

Here. A, and A> are arbitrary constants, and />? * *bt. r, and r> are referred to as characteristic roots. and (/&¥) is the solution to the characteristic or auxiliary equation: r* + bxr + b2 = 0. Sec Examples 1 to 4 and Problems 18.1 to 18.11. 20.9, 20.10. 20.13. 20.14 and 20.16 to 20.20.

EXAMPLE 1. The particular integral for each of the following equations

(1) /(/)-5y'(/) + 4v</)»2 (2) y"(0 + 3y'(0 = 12 (3) f(t) = 16 is found as shown below.

EXAMPLE 2. The complementary Junctions for equations (1) and (2) in Example 1 arc calculated below Equation (3) will be treated in Example 9.

Substituting in (183a) and (/fl.Jft). and finally in (18 3).

^.ZlimzM.lf.O. -3 Thus, yt » + Aje'* = A, + A,r(18 6a)

EXAMPLE 3. The general solution of a differential equation is composed of the complementary function and the particular integral (Section 16.2). that is, y(f) = yc+ yp. As applied to the equation* in Example !.

For (1). from (186) and (185). y(t) - A,S + A,f* + 1 (187)

For (2). front U86a) and (185a). y(t) = A, + A^"* -»-4/ (18.7a)

EXAMPLE 4. The definite solution for (1) in Example 3 is calculated below. Avsume >(0) = 5i and /(0>-ll.

Evaluating (18.8) and (18Ma) at t = 0. and setting y(0) = 5; and y'(0) = II from the initial conditions.

y(0)~ + + 5j thus A,+A2 = 5 y'(0) = Ai«* + 4A}e*<0) = 11 thus A,+4A?=11 Solving simultaneously. A, = 3 and A; - 2. Substituting in (187).

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