Laplace Transformations
Application to Differential Equations
Introduction
A transformation operates on functions to produce other
functions.
Examples:
Differentiation: D[ f ( x)] = f ( x)
x
Integration: I [ f ( x)] =
f (t )dt
0
Linear Transformation: A transformation T of functions is said
to be linear if
T [ f ( x ) + g ( x ) ] = T [ f ( x ) ] + T [ g ( x )]
For functions f(x) and g(x) and all constants and .
Integral Transformation: Consider a function f(x) defined on
a finite or infinite interval a x b and choose a fixed function
K(p, x) of the variable x and a parameter p, then the general
integral transformation is
b
T [ f ( x)] = K ( p, x) f ( x)dx = F ( p )
a
Laplace Transformation
Function K(p, x) is called kernel of the transformation T, and it
is clear that T is linear regardless of the nature of K.
When a = 0, b = and K(p, x) = e-px, in (1), we get a special
case that is known as Laplace Transformation L defined by
L[ f ( x)] = e px f ( x)dx = F ( p )
0
Laplace Transformation
The Laplace transform is named after mathematician and
astronomer Pierre-Simon Laplace, who used a similar
transform in his work on probability theory
x ( x)ds
s
Laplace transforms are used in a lot of engineering
applications and is a very useful tool. It is useful in solving
differential and integral equations.
The Laplace Transform
An important point to remember:
f ( x) F ( p)
The above is a statement that f(x) and F(p) are
transform pairs. What this means is that for
each f(x) there is a unique F(p) and for each F(p)
there is a unique f(x).
Laplace Transformation
Show that Laplace transformation is a linear transformation?
Find the Laplace transformation of the following functions:
f ( x) = 1
f ( x) = x n
f ( x) = e ax
f ( x) = sin ax
Transform Pairs:
f(x)
F(p)
1
p
1
eax
p a
f (t )
F ( s)
______________________________
1 ______
x
p2
xn
s in a x
cos ax
n!
p n +1
a
p2 + a2
p
p2 + a2
Inverse Laplace Transformation
If F(p) represent the Laplace transform of a function f(t)
i.e. L[f(x)] = F(p)
we then say f(x) is the inverse Laplace transform of F(p)
and we write
f(x) = L-1{F(p)}
Laplace Transformation: Remarks on Theory
The integral
f ( x)dx
0
converges if the limit
b
lim f ( x)dx
b
exist, and in this case the value of the limit is the value of the
integral.
Laplace Transformation: Remarks on Theory
If f(x) is a given function defined for x 0, convergence of
f ( x)dx requires the integral f ( x)dx must exists for each finite b > 0.
To guarantee this, it suffices to assume that f(x) is continuous or at
least piecewise continuous.
If f(x) is piecewise continuous for x 0, then the only remaining
threat to the existence of the Laplace transform
F ( p) = e px f ( x)dx
0
is the behavior of its integrand e px f ( x) for large x.
To make sure that f(x) does not grow more rapidly, assume
that f(x) is of exponential order:
i.e. there exists constants M and c s.t. | f ( x) | Mecx .
If f(x) satisfies | f ( x) | Mecx x 0
then
| e px f ( x) | Me ( p c ) x
Since the integral on the right converges for p>c, the Laplace
transform of f(x) converges absolutely and hence converges
for p > c.
Theorem: If f(x) is a function which is piecewise continuous
Theorem:
on every finite interval in the range x 0 and satisfies
| f ( x) | Mecx .
and for some constants c and M, then the Laplace transform
of f(x) exists for all p > c.
| f ( x) | MeCx x 0
Laplace Transformation : Remarks on Theory
Remark: Conditions in the theorem are sufficient but not
necessary for the existence of Laplace transform.
Example
f ( x) = x 1/ 2
L[ f ( x)] =
Problems:
(a)
(b)
Show that F ( p ) 0 as p whenever F(p) exists.
Find the Laplace transform of
f(x) = [x], where [x] denotes the greatest integer x.
Applications to Differential Equations
Suppose we wish to find the particular solution
y ''+ ay '+ by = f ( x )
y (0) = y 0 , y '(0) = y 0
where a, b, y0, y0 are known constants.
We use the methods discussed in the previous sections.
Laplace transform gives an alternate way of solving the
above problem with several advantages.
Applications to Differential Equations
Let us apply the Laplace transform to both sides:
L[ y ''+ ay '+ by ] = L[ f ( x)]
L[ y] + aL[ y] + bL[ y ] = L[ f ( x)].
by the linearity if L.
Next, express the LHS in terms of L[y].
Laplace transform of derivatives
L[ y] = pL[ y] y(0),
L[ y] = p2 L[ y] py(0) y(0),
......
L[ y( n) ] = pn L[ y] pn1 y(0) pn2 y(0) y( n1) (0).
Using these transformations we get
p 2 L[ y ] py0 y0 + a pL[ y ] ay0 + bL[ y ] = L[ f ( x )].
Solving for L[y] yields
L[ f ( x )] + ( p + a ) y0 + y0
L[ y ] =
p 2 + ap + b
Now taking the inverse Laplace transform to get the solution
y(x) of the differential equation.
Note: The above mentioned procedure is particularly suited
to problems in which the function f(x) is discontinuous.
Problems:
Find the solution of the IVPs
(i) y + 4 y = 4x, y(0) = 1, y(0) = 5,
2
(ii) y + y = 3x , y(0) = 0, y(0) = 1,
x
(iii) y + 2 y + 5y = 3e sin x, y(0) = 0, y(0) = 3.
Useful Rules
First translation or shift rule:
If L[f(x)] = F(p) when p > c, then
L[eax f(x)] = F(p - a),
Equivalently,
L-1[F(p a)]=eax f(x).
Used to find transforms of product of the form eax f(x), when
F(p) is known.
Derivatives and Integrals of Laplace
Transformation
If L[f(x)] = F(p), then
( a ) L [ x n f ( x )] = ( 1) n F ( n ) ( p )
x
F ( p)
( b ) L f ( x ) dx =
p
0
f ( x)
(c ) L
=
f ( x)
exists.
F ( p ) dp provided lim
x
x0
The formulas in the previous slide are useful in solving
differential equations with variable coefficients.
Can be used to evaluate Laplace transforms of some
typical functions
Can be used to evaluate integrals that are difficult to
handle by any other methods.
Problems:
(i) Solve the Bessels equation
xy + y + xy = 0, y (0) = 1.
(ii) Find the Laplace transform of f ( x ) = x 3 / 2 .
(iii) Show that
cos xt
-x
0 1 + t 2 dt = 2 e .
Laplace Transformation: Convolutions
Let L[f(x)] = F(p) and L[g(x)] = G(p).
Then,
L 1 [ F ( p ) G ( p ) ] =
f ( x t ) g (t ) d t.
Laplace Transformation: Convolutions
The integral on the RHS is called convolution of the
functions f(x) and g(x). The result of called as Convolution
Theorem.
Is used to find the inverse transforms and solving integral
equations.
Application to Integral Equation
If f(x) and k(x) are known function then the equation
x
y ( x) = f ( x) k ( x t ) y (t )dt
0
is called integral equation.
and therefore, we have the solution given by the inverse of
L[ f ( x)]
L[ y ( x)] =
1 + L[k ( x)]
Problems:
Find the inverse Laplace transform of
1
(i ) F ( p ) = 2
p ( p 2 + 1)
1
( ii ) G ( p ) =
( p 2 + a 2 )2
Solve the following integral equation using Laplace transform
x
y(x) = x3 +
s in ( x t ) y (t ) d t
0
Some more problems:
If f(x) is a periodic function with period a show that
1
L [ f ( x )] =
1 e ap
px
e
f ( x)dx
Solve the following differential equation using Laplace
transform
x y + ( 3 x 1) y ( 4 x + 9 ) y = 0 , y ( 0 ) = 0 .
