Laplace Transform

1. A.Sujatha, M.Sc., M.Phil.,PGDCA.,
M.Mohana Malar, M.Sc.,M.Phil.,
T.Kalaiselvi, M.Sc.,M.Phil.,

2. The general second order linear
differential equation is of the form
𝑑2 𝑦
𝑑𝑥2 + 𝑃
𝑑𝑦
𝑑𝑥
+ 𝑄 = R, where P,Q and R are
functions of x.

3. Given equation is of the form
𝐴𝑦" + 𝐵𝑦′ + 𝑐𝑦 = 𝑔(𝑡)
Suppose we know the general solution to the
homogeneous equation.
𝐴𝑦" + 𝐵𝑦′ + 𝑐𝑦 = 0
Which takes the form 𝑦 = 𝐶1 𝑦1 + 𝐶2 𝑦2

4. Classifications of integrals
Let the partial differential equation be
F(x,y,z,p,q)=0.
Let the solution of this be 𝜑(x,y,z,a,b)=0 where a and
b are arbitrary constants.
Singular integral
The eliminant of a and b between 𝜑(x,y,z,a,b)=0;
𝛿𝜑
𝛿𝑎
= 0 and
𝛿𝜑
𝛿𝑏
= 0 when it exists is called the singular
integral.

5. Eliminate a and b from z=(x+a)(y+b).
Soln:
Differentiating with respect to x and y partially,
P=y+b and q=x+a
Eliminating a and b, we get Z=pq.
Eliminate the arbitrary function from 𝒛 = 𝒇(𝒙 𝟐
+ 𝒚 𝟐
)
Soln:
Differentiating with respect to x and y partially,
𝑃 = 𝑓′
𝑥2
+ 𝑦2
2𝑥 and 𝑞 = 𝑓′
𝑥2
+ 𝑦2
2𝑦
Eliminating 𝑓′
𝑥2
+ 𝑦2
, we get
py=qx

6. To solve the linear equation Pp+Qq=R is as follows:
Write down the subsidiary equations
𝑑𝑥
𝑃
=
𝑑𝑦
𝑄
=
𝑑𝑧
𝑅
. Let the
two independent integrals of these ordinary differential
equations be u=a and v=b. Then the solution of the given
equation is 𝜑 𝑢, 𝑣 = 0, where 𝜑 is an arbitrary function and is
called the general integral of Lagrange’s Linear equations.
Cor 1: This equations can be extended to the n independent
variables. Then 𝜑 𝑢1, 𝑢2 … … , 𝑢 𝑛 = 0 is the solution.
Cor 2: Either u=a or v=b involves z it is an integral of the
differential equation. 𝜑 𝑢, 𝑣 = 0 can be written as 𝑢 = 𝑓 v , f
is arbitrary. We take 𝑓 𝑣 = 𝑎𝑣0
, where a is an arbitrary constnt
thus the solution reduces to u=a.

7. Definition
If a function f(t) is defined for all
positive values of the variables t and if
exists and is equal
to F(s), then F(s) is called the Laplace
transforms of f(t) and is denoted by the
symbol L{f(t)}.
*
dttfe st



0
)(

8. *
Note
dttfe st



0
)(
.0)( 

sFLts

9. (i) L{f(t) + (t) = L{f(t)} + L{ (t)}.
(ii) L{cf(t)} = cL{f(t)}.
(iii) L{f’(t)} = sL{f(t)} – f(0).
(iv) L{f”(t)} = L{f(t)}– sf(0) –
f’(0).
* Results
 
2
s

10. *Initial Value Theorem and Final
Value Theorem
)(lim)(lim
0
ssFtf
st 

)(lim)(lim
0
ssFtf
st 


11. Results
(i)
(ii)
Inverse Laplace Transforms
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