Begin2.DVI

so that the differential equation in the t-domain becomes an algebraic equation in

the s-domain. The resulting algebraic equation is

sY (s)−1 = α Y (s)

One can now solve this algebraic equation for the transform function Y(s)to obtain

Y(s) =

1

s−α

Using table lookup one finds the inverse Laplace transform

y(t) = L−^1 {Y(s)}=L−^1

{

1

s−α

}

=eαt

One can verify the correctness of the solution by showing the function y(t) = eαt

satisfies the given differential equation and given initial condition.

Warning—The Laplace transform technique for solving differential equations only

works on linear differential equations. It is not applicable in dealing with nonlinear

differential equations. Also note that when dealing with more difficult linear equa-

tions one needs to develop more advanced methods for obtaining an inverse Laplace

transform.

Introduction to Complex Variable Theory

Consider the figure 12-10, where Srepresents a nonempty set of points in the

z =x+i y complex z-plane, where iis an imaginary unit with the property that

i^2 =− 1 .If there exists a rule f which assigns to each value z=x+i y belonging to

S, one and only one complex number ω=u+i v, then the correspondence is called a

function or mapping of the point zto the point ωand this correspondence is denoted

using the notation

ω=f(z) = f(x+i y) = u+i v =u(x, y ) + i v(x, y )

Here ω=u+i v is the image point of z=x+i y and is represented in a plane called the

ω−plane. Functions of a complex variable ω=f(z)are represented as mappings from

the z−plane to the ω−plane as illustrated in the figure 12-10. Note that if Sdenotes

a region in the z−plane and f(z)is a single-valued, then the image of Sunder the

mapping ω=f(z)is the region S′in the ω−plane. The boundary curve Cof Sin the

z−plane has the image curve C′in the ω−plane.