Laplace Transforms
02.
L {K1 f1 (t) + K2 f2 (t)} (s) = K1 L {f1 (t)} (s) + K2 L {f2 (t)} (s)
n
o
L y (n) (t) (s) = sn L {y(t)} (s) − sn−1 y(0) − sn−2 y 0 (0) − · · · − y (n−1) (0) n = 1, 2, 3, · · ·
03.
L ea t f (t) (s) = L {f (t)} s − a
04.
L {ua (t) f (t)} (s) = L f t + a (s) e−a s
01.
05.
06.
08.
10.
12.
14.
16.
Z T
1
L {f (t)} (s) =
f (t) e−s t dt, f (t + T ) = f (t)
1 − e−T s 0
Z +∞
d
f (t)
L {t f (t)} (s) = −
L {f (t)} (s)
07. L
(s) =
L {f (t)} (r) dr
ds
t
s
L {tn } (s) =
n!
sn+1
,
n = 0, 1, 2, · · ·
s
s2 + b2
b
L sin b t (s) = 2
s + b2
e−a s
L {ua (t)} (s) =
rs
1
π
L √ (s) =
s
t
L cos b t (s) =
n!
n+1 , n = 0, 1, 2
s−a
s−a
cos b t (s) =
(s − a)2 + b2
b
sin b t (s) =
(s − a)2 + b2
1
(s) =
s−a
09.
L ea t tn (s) =
11.
L ea t
13.
L ea t
15.
L ea t
Inverse Laplace Transforms
01.
L−1 {K1 F1 (s) + K2 F2 (s)} (t) = K1 L−1 {F1 (s)} (t) + K2 L−1 {F2 (s)} (t)
02.
L−1 F s − a (t) = ea t L−1 {F (s)} (t)
03.
L−1 F (s) e−a s
04.
L−1 {F 0 (s)} (t) = −t L−1 {F (s)} (t)
05.
L−1
07.
09.
L−1 F s + a (t) = e−a t L−1 {F (s)} (t)
−a s e
−1
−1
(t) = ua (t) L {F (s)} t − a
L
(t) = ua (t)
s
tn
(t) =
n = 0, 1, 2, · · ·
n+1
n!
s
s
L−1
(t) = cos b t
2 + b2
s
1
1
−1
L
(t) = sin b t
2
2
s +b
b
1
or
06.
08.
10.
1
tn
(t) = ea t
n = 0, 1, 2
n+1
n!
(s − a)
s−a
L−1
(t) = ea t cos b t
2 + b2
(s
−
a)
1
1
−1
L
(t) = ea t sin b t
2
2
(s − a) + b
b
L−1
Trigonometric Identities
1.
4.
7.
cos2 (θ) + sin2 (θ) = 1
2 cos2 (θ) = 1 + cos(2 θ)
sin(θ ± π) = − sin(θ)
2.
5.
8.
cos(2 θ) = cos2 (θ) − sin2 (θ)
2 sin2 (θ) = 1 − cos(2 θ)
cos(θ ± 2 π) = cos(θ)
1
3.
6.
9.
sin(2 θ) = 2 cos(θ) sin(θ)
cos(θ ± π) = − cos(θ)
sin(θ ± 2 π) = sin(θ)
Fourier Series
1.
If f (x) is 2 L-periodic, then its Fourier series is
+∞
π π a0 X
+
an cos n x + bn sin n x
2
L
L
n=1
where
an =
2.
1
L
Z
π f (x) cos n x dx,
L
I
1
L
Z
π f (x) sin n x dx
L
I
and I is an interval of length 2 L
If f (x) is defined on the interval 0 , L , then its Fourier sine series is
+∞
X
π bn sin n x ,
L
n=1
3.
bn =
If f (x) is defined on the interval
where
bn =
2
L
Z
0
L
π f (x) sin n x dx
L
0 , L , then its Fourier cosine series is
+∞
π a0 X
+
an cos n x ,
2
L
n=1
where
2
an =
L
Z
0
L
π f (x) cos n x dx
L
Boundary Value Problems
1.
are given by
2.
n2 π 2
L2
and
y 00 + λ y = 0, 0 < x < L
y 0 (0) = 0, y 0 (L) = 0
n π and yn (x) = cos
x , n = 0, 1, 2, · · ·
L
λn =
n2 π 2
L2
y 00 + λ y = 0, 0 < x < L
y(0) = 0, y 0 (L) = 0
(2 n − 1) π
and yn (x) = sin
x , n = 1, 2, 3, · · ·
2L
The eigenvalues-eigenfunctions of the BVP
are given by
4.
λn =
The eigenvalues-eigenfunctions of the BVP
are given by
3.
y 00 + λ y = 0, 0 < x < L
y(0) = 0, y(L) = 0
n π x , n = 1, 2, 3, · · ·
yn (x) = sin
L
The eigenvalues-eigenfunctions of the BVP
λn =
(2 n − 1)2 π 2
4 L2
y 00 + λ y = 0, 0 < x < L
y 0 (0) = 0, y(L) = 0
(2 n − 1) π
and yn (x) = cos
x , n = 1, 2, 3, · · ·
2L
The eigenvalues-eigenfunctions of the BVP
are given by
λn =
(2 n − 1)2 π 2
4 L2
2