Principles of Mathematics in Operations Research

194 14 Special Transformations

51. y(t)^V(s).


52. Solve the resulting linear algebraic equation, call the solution n(s) the
    formal Laplace transform of y(t).


53. Find the inverse-Laplace transform y(t).
    S4- Verify that y(t) is a solution.

Example 14.2.5 Find the solution to

y'(t) = -4y(t) + f(t); 2/(0) = 0,

where f(t) is the unit step function

and I = [0,oo). Transforming both sides, we have

e~s

sr)(s) - j/(0) = -4n(s) +

s

e~s

sn(s) — -4n(s) H.

At the end of S2, we have n{s) = Sfs+A\ •

1 1/1 1

s(s + 4) 4 \s s + 4/ '

Therefore,

V(s) = -.e

s s + 4

Thus,

^_/0, t<l;

Example 14.2.6 Let us solve

y'(t) = ay(t) + f(t); 2/(0) = 0

such that y'(t) = f(t).
Let us take y'(t) = f(t) then sn(s) — y 0 — (f)(s), where 4>(s) = £ {/(<)}. Thus,

V(s) =2/o- + -<A(s).

s s

We use formula (9) in Table 14-2.

y(t) = 2/o + / f(u) du.

Jo