Laplace Transform of Derivatives
Laplace of the First Derivative:
L L
Laplace of the Second Derivative:
L L
Laplace of the Third Derivative:
L L
Laplace of the nth Derivative:
L L
Solution of Linear Differential Equations with Constants Coefficients
Using Laplace Transformation
1. The given differential equation is transformed into an algebraic
equation called the subsidiary equation.
2. The subsidiary equation is solved by purely algebraic manipulations.
3. The solution of the subsidiary equation is transformed back to the
original function so that it becomes the required solution of the given
differential equation.
Consider the Linear Differential Equation with Constant Coefficients:
a
n
(
x
)
d
n
y
dx
n
+ a
n−1
(
x
)
d
n−1
y
dx + a
n−2
(
x
)
d
n−2
y
dx +...+ a
1
(
x
)
dy
dx + a
0
(
x
)
y = F
(
x
)
a
n
D
n
y + a
n−1
D
n−1
y + a
n−2
D
n−2
y +. . .+ a
1
Dy + a
0
y = F(x)
(
a
n
D
n
+ a
n−1
D
n−1
+ a
n−2
D
n−2
+...+ a
1
D + a
0
)
y = F(x)
With initial conditions:
y
(
0
)
=C
0
y''
(
0
)
=C
2
y '
(
0
)
=C
1
y'''
(
0
)
=C
3
Steps in solving the Differential Equation:
1. Take the Laplace Transform of both sides of the given linear differential
equation.
2. Apply the initial conditions and solve the resulting algebraic equation
to obtain L .
3. Find the inverse transform.
L
−1
L
{
y
(
t
)
}
= y
(
t
)
=F
(
x
)
L
−1
L
{
F
(
s
)
}
= y
(
t
)
=F
(
x
)
{
y
(
t
)
}
−y
(
0
)
{
y '
(
t
)
}
= F
(
s
)
= s
{
y
(
t
)
}
−sy
(
0
)
−y '
(
0
)
¿¿
{
y
(
t
)
}
−s
2
y
(
0
)
−s y '
(
0
)
−y left (0 right )} { ¿
¿¿
{
y
(
t
)
}
−s
n−1
y
(
0
)
−s
n−2
y '
(
0
)
−s
n−3
y left (0 right ) - . . . - y rSup { size 8{ left (n - 1 right )'} } left (0 right )} {¿
{
y
n
(
t
)
}
= F
(
s
)
= s
n
{
y '
(
t
)
}
= F
(
s
)
