MATH 417 ADVANCE ENGG MATHCOLLEGE OF ENGINEERING

COLEGIO SAN AGUSTIN

Topic: Laplace TransformPrepared by: Engr. Vicente Y. Buenconsejo, Jr., ECE

Pierre Simon Marquis De Laplace (1749-1827), great French mathematician develop the theory of potential and contributions to celestial mechanics, astronomy in general, special functions, and probability theory. Napoleon Bonaparte was his student for a year.

3 Main Steps1. The given hard problem is tranmsformed into a simple equation called subsidiary

equation.2. The subsidiary equation is solved by purely algebraic manipulations.3. The solution of the subsidiary equation is transformed back to obtain the solution of the

given problem.

These process is made easier by tables of functions and their transforms, whose roleis similar to that of integral tables of calculus. The switching from operations of calculus to algebraic operations on transforms is called OPERATIONAL CALCULUS. For engineers, Laplace transform method is practically the most important operational method.

Let given function defined for all Multiply by and integrate with respect to from to . We have;

where: = Laplace transform of the original function and will be denoted by L(f).

= original function

(1) = L(f) =

Remember that:depends on depends on

The original function in (1) is called the inverse transform or inverse of and will be denoted by (F).

= (F)Notation:

Original functions are denoted by lowercase lettersTransforms by the same letters in CAPITALS

So that: denotes the transform of

denotes the transform of

Example 1.= 1 when Find .

From (1) by integration:

= L(f) =

=

=

=

=

=

= L(1) = ans.

Example 2. Laplace Transform of the exponential function.= 1 when Find where a is constant.

Solution:

= L(f) =

= L( ) = = =

=

= =

=

= L( ) = ans.

THEOREM 1: Linearity of the transform

PROOF: By definition;

Application of theorem 1:Example 3:

. Find .

Solution:

=

=

=

=

= ans.

Using the table:

Example 4: . Find .

Solution:

ans.

Example 5:. Find .

Solution:

=

= ans.

TRANSFORMS OF DERIVATIVES:Laplace transforms replaces operations of calculus by operations of algebra on

transforms.

“ The differentiation of is replaced by multiplication of by .”

(1) ; is continuous for all where:

= Laplace transform of the first derivative of = Laplace transform of ; also = value of the function at time

And:; substitute in (1)

Similarly;

Or in general;

Example 1:= . Derive .

Solution:1. Take the derivatives

= = = 2

2. Take the Laplace of the highest derivative of .

3. Substitute to equation of differentiation of the highest derivative

From the table

ans. (see table)

Note: Different Process but same answer!!!

Example 2:Derive the Laplace transform of .

Solution:1. Differentiate

2. Take the Laplace of the highest derivative of .

But: = ; constant

3. Substitute to equation of differentiation of the highest derivative

ans. (see table)

Assignment:1. Derive using transforms of derivative.2.

INITIAL VALUE PROBLEMS:

Given the Differential Equation:(5)

where:

= input (driving force) applied to the mechanical system = output (response of the system)

1st Step: Transform (5) by means of (1) and (2) writing &

this is the subsidiary equationCollecting Y terms we have;

2nd Step: Solve the subsidiary equation algebraically for Y. Multiply the subsidiary equation by:

(6)

(7)If

Note that Q depends only on a & b, neither on r(t) or initial condition.

3rd Step: Reduce (7) usually by partial fraction, to a sum of terms whose inverse can be found from the table so that;

of (5) is obtained.

Example 1:Solve: ; ;

Solution:1. From the table and (2) get the subsidiary equation

2. The transfer function is and (7) becomes;

3. From this expression, find from table.

ans.

Diagram:

Given Problem Subsidiary Equation

t space s space

Solution of Subsidiary Solution of the Given Equation Problem

t space s space

Practice Problems:1. 2. 3. 4.

LAPLACE TRANSFORM OF THE INTEGRAL OF A FUNCTION

Theorem:“Integration of a function corresponds to division of its transform by s”

;Or if we take the inverse transform

where:

Example:

Find

Solution:From the table we have:

Note that:

From this and Theorem 3 we have:

Note from table:

Then;

ans.

Example 2: Derive the formula 20 in table using the transform of Integrals

Find

Solution:

From the table we have:

ans.

Practice Problems:

1.

2.