The document discusses the Laplace transform and its uses. The Laplace transform converts a differential equation into an algebraic equation, making it easier to solve. It allows one to directly find the particular solution of a differential equation without first finding the general solution. The Laplace transform also allows solving nonhomogeneous equations directly without first solving the corresponding homogeneous equation. It can also be used to find solutions to problems with discontinuous driving forces. The Laplace transform of a function f(t) is defined as the integral from 0 to infinity of f(t)e^-st dt. It transforms the function from depending on t to depending on s.

1. YASH PATEL
http://alltypeim.blogspot.in/

2. INTRODUCTION
• The word transform itself Indicates about the
conversion one form to another form. The
Laplace transformation is a very versatile tool
for the solving differential equations as it
transforms differential equations to algebraic
equation.
• we know algebraic equation are comparatively
more easy to solve.
http://alltypeim.blogspot.in/

3. Uses of Laplace Transforms
1. Particular solution of a differential equation is directly
obtained without first determining general solution.
2. Nonhomogeneous differential equation can be solved
directly without first solving the corresponding
homogeneous equation.
3. Finding solution of problems were mechanical and
electrical driving force has discontinuities , is
impulsive or is a complicated periodic function, not
merely sine and cosine or step function.
4. Finding solution of system of O.D.Es , P.D.Es and
integral equations.
http://alltypeim.blogspot.in/

4. Laplace Transform
 Let f be a function defined for t  0, and satisfies certain
conditions to be named later.
 The Laplace Transform of f is defined as
 Thus the kernel function is K(s,t) = e-st.
 Here L is known as Laplace transform operator
 The original given f(t) is known as determining function which
depends on t, while the new function F(s) to be determined is
called generating function and depends only on s.
  



0
)()()( dttfesFtfL st
http://alltypeim.blogspot.in/

5. Example
 
as
as
as
e
dte
dteeeL
btas
b
b
tas
b
atstat















,
1
lim
lim
0
)(
0
)(
0
 Let f (t) = eat for t  0. Then the Laplace transform F(s) of f is:
http://alltypeim.blogspot.in/

6. Piecewise Continuous Function
A function f is piecewise continuous on an interval [a, b] if
this interval can be partitioned by a finite number of points
a = t0 < t1 < … < tn = b such that
(1) f is continuous on each (tk, tk+1)
In other words, f is piecewise continuous on [a, b] if it is
continuous there except for a finite number of jump
discontinuities.
nktf
nktf
k
k
tt
tt
,,1,)(lim)3(
1,,0,)(lim)2(
1









http://alltypeim.blogspot.in/

7. Example
 Consider the following piecewise-defined function f.
 From this definition of f, and from the graph of f below, we
see that f is not piecewise continuous on [0, 3].
 










32,4
21,2
10,1
)(
1
2
t
tt
tt
tf
http://alltypeim.blogspot.in/

8. Formulae
•Some Laplace Transforms
http://alltypeim.blogspot.in/

9. Laplace transforms of ODEs
• Equation with initial conditions
• Laplace transform is linear
• Apply derivative formula
• Rearrange
• Take the inverse
http://alltypeim.blogspot.in/

10. Laplace transform of PDEs
• Laplace transform in two
variables (always taken with
respect to time variable, t):Inverse laplace of a 2 dimensional PDE:
Can be used for any dimension PDE:
The Transform reduces dimension by “1”:
•ODEs reduce to algebraic equations
•PDEs reduce to either an ODE (if original equation
dimension 2) or another PDE (if original equation
dimension >2)
http://alltypeim.blogspot.in/

11. http://alltypeim.blogspot.in/