Download Differential Equations Formula Sheet Trigonometric identities • sin 2

Differential Equations Formula Sheet
Trigonometric identities
• sin2 x + cos2 x = 1
• sin 2x = 2 sin x cos x
• cos 2x = cos2 x − sin2 x = 2 cos2 x − 1 = 1 − 2 sin2 x
• sec2 x = 1 + tan2 x
• csc2 x = 1 + cot2 x
1
• sin x cos y = [sin(x − y) + sin(x + y)]
2
1
• sin x sin y = [cos(x − y) − cos(x + y)]
2
1
• cos x cos y = [cos(x − y) + cos(x + y)]
2
2t
1 − t2
2t
x
,
cos
x
=
,
and
tan
x
=
• if t = tan then sin x =
2
1 + t2
1 + t2
1 − t2
Hyperbolic identities
• cosh2 x − sinh2 x = 1
• sinh 2x = 2 sinh x cosh x
• cosh 2x = cosh2 x + sinh2 x = 2 cosh2 x − 1 = 2 sinh2 x + 1
• sech2 x = 1 − tanh2 x
• csch2 x = coth2 x − 1
1
• sinh x cosh y = [sinh(x + y) + sinh(x − y)]
2
1
• sinh x sinh y = [cosh(x + y) − cosh(x − y)]
2
1
• cosh x cosh y = [cosh(x + y) + cosh(x − y)]
2
1
Derivatives of trigonometric and hyperbolic functions
d
[sin x] = cos x
dx
d
[cos x] = − sin x
dx
d
[tan x] = sec2 x
dx
d
[cot x] = − csc2 x
dx
d
[sec x] = sec x tan x
dx
d
[csc x] = − csc x cot x
dx
d
1
[tan−1 x] =
dx
1 + x2
d
[sinh x] = cosh x
dx
d
[cosh x] = sinh x
dx
d
[tanh x] = sech2 x
dx
d
[coth x] = −csch2 x
dx
d
[sechx] = −sechx tanh x
dx
d
[cschx] = −cschx coth x
dx
Exponents and logarithms
• y = ax
•
⇔
x = loga y
d x
[a ] = ax · ln a and
dx
and y = ex
d
dx [loga x]
=
⇔
x = ln y
1
x ln a
Matrices
a11 a12
Let A =
.
a21 a22
1
a22 −a12
−1
•A =
.
det A −a21 a11
det A1
det A2
• Cramer’s Rule: the solution of Ax = b is given by x1 =
, x2 =
det A
det A
b a
a11 b1
where A1 = 1 12 , A2 =
.
b2 a22
a21 b2
2
Sums and series
•
n
X
k=
k=1
•
n
X
k2 =
k=1
•
n
X
k=1
n(n + 1)
2
n(n + 1)(2n + 1)
6
n(n + 1)
k =
2
3
2
• Geometric progression:
Geometric series:
∞
X
n−1
X
k=0
rk =
k=0
• Maclaurin series:
k!
∞
X
f (k) (a)
k=0
1
when
1−r
∞
X
f (k) (0)
k=0
• Taylor series:
rk = 1 + r + r2 + · · · + rn−1 =
k!
1 − rn
, r 6= 1
1−r
|r| < 1
xk
(x − a)k
∞
a0 X
• Fourier series for f (x) on [−π, π]:
+
(an cos nx + bn sin nx)
2
n=1
Z π
Z
1 π
1
f (x) cos nx dx, bn =
f (x) sin nx dx.
where an =
π −π
π −π
3
Transforms
Z
∞
f (x)e−sx dt = F (s)
• Laplace transform: L{f (x)} =
0
L{1} =
L{tn } =
1
s
n!
, n a positive integer
sn+1
1
L{eax } =
s−a
s
L{cos ax} = 2
s + a2
a
L{sin ax} = 2
s + a2
L{f 0 (x)} = sF (s) − f (0)
L{f (n) (x} = sn F (s) − sn−1 f (0) − sn−2 f 0 (0) − · · · − sf (n−2) (0) − f (n−1) (0)
Z ∞
• Fourier transform: F{f (x)} =
f (x)eiαx dt = F (α)
−∞
0
F{f (x)} = −iαF (α)
1
Inverse Fourier transform: F −1 {F (α)} =
2π
4
Z
∞
−∞
F (α)e−iαx dα = f (x).