The document discusses several topics in mathematics including: 1. Differential equations in the form of Ay" + By′ + cy = g(t) and their homogeneous solutions. 2. Classifications of integrals including singular integrals which are found by eliminating arbitrary constants from an integral equation. 3. Solving Lagrange's linear equations by writing subsidiary equations and finding integrals to get the general integral solution in the form φ(u,v)=0. 4. The Laplace transform of a function f(t), denoted by L{f(t)}, which is defined as the integral of f(t)e^-st from 0 to infinity if it exists. Some properties of the Laplace transform are

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)}.
*
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8. *
Note
dttfe st



0
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
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
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0
ssFtf
st 

)(lim)(lim
0
ssFtf
st 


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