1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

6.2 Partial Fractions and Convolutions 369

a.^1

(s^2 +ω^2 )^2

;

c. s

2

(s^2 +ω^2 )^2

;

b. s

(s^2 +ω^2 )^2

;

d. s

3

(s^2 +ω^2 )^2

6.2 Partial Fractions and Convolutions

Because of the formula for the transform of derivatives, the Laplace transform
finds important application to linear differential equations with constant co-
efficients, subject to initial conditions. In order to solve the simple problem

u′+au= 0 , u( 0 )= 1 ,

we transform the entire equation, obtaining

L(u′)+aL(u)= 0

or

sU− 1 +aU= 0 ,

whereU=L(u). The derivative has been “transformed out,” andUis deter-
mined by simple algebra to be

U(s)=

1

s+a.

By consulting Table 2 we find thatu(t)=e−at.
Equationsofhigherordercanbesolvedinthesameway.Whentrans-
formed, the problem

u′′+ω^2 u= 0 , u( 0 )= 1 , u′( 0 )= 0

becomes

s^2 U−s· 1 − 0 +ω^2 U= 0.

Note how both initial conditions have been incorporated into this one equa-
tion. Now we solve the transformed equation algebraically to find

U(s)=

s

s^2 +ω^2 ,

the transform of cos(ωt)=u(t).