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Trig Cheat Sheet
Definition of the Trig Functions Right triangle definition For this definition we assume that
02
! " < < or 0 90" ° < < ° .
oppositesin
ypotenuse" =
ypotenusecsc
opposite" =
adjacentcos
ypotenuse" =
ypotenusesec
adjacent" =
oppositetan
adjacent" =
adjacentcot
opposite" =
Unit circle definition
For this definition " is any angle.
sin1
y y" = =
1csc
y" =
cos1
x x" = =
1sec
x" =
tan y
x" = cot
x
y" =
Facts and PropertiesDomain
The domain is all the values of " thatcan be plugged into the function.
sin " , " can be any angle
cos" , " can be any angle
an" ,1
, 0, 1, 2,2
n n" ! ! "
# + = ± ±$ %& '
…
csc" , , 0, 1, 2,n n" ! # = ± ± …
sec" ,1
, 0, 1, 2,2
n n" ! ! "
# + = ± ±$ %& '
…
cot" , , 0, 1, 2,n n" ! # = ± ± …
Range The range is all possible values to getout of the function.
1 sin 1" ( ) ) csc 1 and csc 1" " * ) (
1 cos 1" ( ) ) sec 1 andsec 1" " * ) (
tan" (+ < < + cot" (+ < < +
PeriodThe period of a function is the number,
T , such that ( ) ( ) f T f " " + = . So, if #
is a fixed number and " is any angle we
have the following periods.
( )sin #" , 2
T !
# =
( )cos #" , 2
T !
# =
( )tan #" , T !
# =
( )csc #" , 2
T !
# =
( )sec #" , 2
T !
# =
( )cot #" , T !
# =
"
adjacent
oppositehypotenuse
x
y
, x y
"
x
y 1
Formulas and IdentitiesTangent and Cotangent Identities
sin costan cot
cos sin
! ! ! !
! ! = =
Reciprocal Identities
1 1csc sin
sin csc
1 1sec cos
cos sec
1 1cot tanan cot
! ! ! !
! ! ! !
! ! ! !
= =
= =
= =
Pythagorean Identities 2 2
2 2
2 2
sin cos 1
an 1 sec
1 cot csc
! !
! !
! !
+ =
+ =
+ =
Even/Odd Formulas
( ) ( )
( ) ( )
( ) ( )
sin sin csc csc
cos cos sec sec
an tan cot cot
! ! ! !
! ! ! !
! ! ! !
! = ! ! = !
! = ! =
! = ! ! = !
Periodic FormulasIf n is an integer.
( ) ( )
( ) ( )
( ) ( )
sin 2 sin csc 2 csc
cos 2 cos sec 2 sec
an tan cot cot
n n
n n
n n
! " ! ! " !
! " ! ! " !
! " ! ! " !
+ = + =
+ = + =
+ = + =
Double Angle Formulas
( )
( )
( )
2 2
2
2
2
sin 2 2sin cos
cos 2 cos sin
2cos 1
1 2sin
2tantan 2
1 tan
! ! !
! ! !
!
!
! !
!
=
= !
= !
= !
=!
Degrees to Radians Formulas
If x is an angle in degrees and t is anangle in radians then
180and
180 180
t x t t x
x
" "
" = " = =
Half Angle Formulas
( )( )
( )( )
( )
( )
2
2
2
1sin 1 cos 2
2
1cos 1 cos 2
2
1 cos 2tan
1 cos 2
! !
! !
! !
!
= !
= +
!=
+
Sum and Difference Formulas
( )
( )
( )
sin sin cos cos sin
cos cos cos sin sin
tan tantan
1 tan tan
# $ # $ # $
# $ # $ # $
# $ # $
# $
± = ±
± =
±± =
!
!
Product to Sum Formulas
( ) ( )
( ) ( )
( ) ( )
( ) ( )
1sin sin cos cos
2
1cos cos cos cos
2
1sin cos sin sin
2
1cos sin sin sin2
# $ # $ # $
# $ # $ # $
# $ # $ # $
# $ # $ # $
= ! ! +# $% &
= ! + +# $% &
= + + !# $% &
= + ! !# $% &
Sum to Product Formulas
sin sin 2sin cos2 2
sin sin 2cos sin2 2
cos cos 2cos cos2 2
cos cos 2sin sin2 2
# $ # $ # $
# $ # $ # $
# $ # $ # $
# $ # $ # $
+ !' ( ' (+ = ) * ) *
+ , + ,
+ !' ( ' (! = ) * ) *
+ , + ,
+ !' ( ' (+ = ) * ) *
+ , + ,
+ !' ( ' (! = ! ) * ) *
+ , + , Cofunction Formulas
sin cos cos sin2 2
csc sec sec csc2 2
an cot cot tan2 2
" " ! ! ! !
" " ! ! ! !
" " ! ! ! !
' ( ' (! = ! =) * ) *
+ , + ,
' ( ' (! = ! =) * ) *
+ , + ,
' ( ' (! = ! =) * ) *
+ , + ,
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Unit Circle
or any ordered pair on the unit circle ( ), x y : cos x! = and sin y! =
xample
5 1 5 3cos sin
3 2 3 2
" " ! " ! "= = #$ % $ %
& ' & '
3
"
4
"
6
"
2 2,
2 2
! "$ %$ %& '
3 1,
2 2
! "$ %$ %& '
1 3,
2 2
! "$ %$ %& '
60°
45°
30°
2
3
"
3
4
"
5
6
"
7
6
"
5
4
"
4
3
"
11
6
"
7
4
"
5
3
"
2
"
"
3
2
"
0
2"
1 3,
2 2
! "#$ %
& '
2 2,
2 2
! "#$ %
& '
3 1,
2 2! "#$ %& '
3 1,
2 2
! "# #$ %
& '
2 2,
2 2
! "# #$ %
& '
1 3,
2 2
! "# #$ %
& '
3 1,
2 2
! "#$ %
& '
2 2,
2 2
! "#$ %
& '
1 3,
2 2
! "#$ %
& '
( )0,1
( )0, 1#
( )1,0#
90°
120°
135°
150°
180°
210°
225°
240° 270°
300°
315°
330°
360°
0°
x
(
)1,0
Inverse Trig FunctionsDefinition
1
1
1
sin is equivalent to sin
cos is equivalent to cos
an is equivalent to tan
y x x y
y x x y
y x x y
!
!
!
= =
= =
= =
Domain and Range Function Domain Range
1sin y x!= 1 1 x! " "
2 2 y! ! ! " "
1cos y x!
= 1 1 x! " " 0 y ! " "
1tan x!
= x!# < < # 2 2
y! !
! < <
Inverse Properties
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
1 1
1 1
1 1
cos cos cos cos
sin sin sin sin
tan tan tan tan
x x
x x
x x
" "
" "
" "
! !
! !
! !
= =
= =
= =
Alternate Notation 1
1
1
sin arcsincos arccos
an arctan
x x x x
x x
!
!
!
=
=
=
Law of Sines, Cosines and Tangents
Law of Sines
sin sin sin
a b c
# $ % = =
Law of Cosines 2 2 2
2 2 2
2 2 2
2 cos
2 cos
2 cos
a b c bc
b a c ac
c a b ab
#
$
%
= + !
= + !
= + !
Mollweide’s Formula
( )12
12
cos
sin
a b
c
# $
%
!+=
Law of Tangents
( )( )
( )
( )
( )
( )
12
12
12
12
12
12
tan
tan
tan
tan
tan
tan
a b
a b
b c
b c
a c
a c
# $
# $
$ %
$ %
# %
# %
!!=
+ +
!!=
+ +
!!=
+ +
c a
b
#
$
%
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Common Derivatives and Integrals
DerivativesBasic Properties/Formulas/Rules
( )( ) ( )d
cf x cf xdx
!= , c is any constant. ( ) ( )( ) ( ) ( ) f x g x f x g x! ! !± = ±
( ) 1n nd x nx
dx
"= , n is any number. ( ) 0
d c
dx= , c is any constant.
( ) f g f g f g ! ! != + – (Product Rule) 2 f f g f g g g
!
! !# $ "=% &' (
– (Quotient Rule)
( )( )( ) ( )( ) ( )d
f g x f g x g xdx
! != (Chain Rule)
( )( ) ( ) ( ) g x g xd g x
dx!=e e ( )( )
( )
( )ln
g xd g x
dx g x
!=
Common Derivatives
Polynomials
( ) 0d
cdx
= ( ) 1d
xdx
= ( )d
cx cdx
= ( ) 1n nd x nx
dx
"= ( ) 1n nd
cx ncxdx
"=
Trig Functions
( )sin cosd
x xdx
= ( )cos sind
x xdx
= " ( ) 2tan secd
x xdx
=
( )sec sec tand
x x xdx
= ( )csc csc cotd
x x xdx
= " ( ) 2cot cscd
x xdx
= "
Inverse Trig Functions
( )1
2
1sin
1
d x
dx x
"=
" ( )1
2
1cos
1
d x
dx x
"= "
" ( )1
2
1tan
1
d x
dx x
"=
+
( )1
2
1sec
1
d x
dx x x
"=
" ( )1
2
1csc
1
d x
dx x x
"= "
" ( )1
2
1cot
1
d x
dx x
"= "
+
Exponential/Logarithm Functions
( ) ( )ln x xd a a a
dx= ( ) x xd
dx=e e
( )( )1
ln , 0d
x xdx x
= > ( )1
ln , 0d
x xdx x
= ) ( )( )1
log , 0ln
a
d x x
dx x a= >
Hyperbolic Trig Functions
( )sinh coshd
x xdx
= ( )cosh sinhd
x xdx
= ( ) 2tanh sechd
x xdx
=
( )sech sech tanhd
x x xdx
= " ( )csch csch cothd
x x xdx
= " ( ) 2coth cschd
x xdx
= "
Common Derivatives and Integrals
IntegralsBasic Properties/Formulas/Rules
( ) ( )cf x dx c f x dx=* * , c is a constant. ( ) ( ) ( ) ( ) x g x dx f x dx g x dx± = ±* * *
( ) ( ) ( ) ( )b b
aa x dx F x F b F a= = "* where ( ) ( ) F x f x dx= *
( ) ( )b b
a acf x dx c f x dx=* * , c is a constant. ( ) ( ) ( ) ( )
b b b
a a a f x g x dx f x dx g x dx± = ±* * *
( ) 0a
a f x dx =* ( ) ( )
b a
a b f x dx f x dx= "* *
( ) ( ) ( )b c b
a a c x dx f x dx f x dx= +* * * ( )
b
ac dx c b a= "*
If ( ) 0 f x + on a x b, , then ( ) 0b
a f x dx +*
If ( ) ( ) f x g x+ on a x b, , then ( ) ( )b b
a a f x dx g x dx+* *
Common Integrals Polynomials
dx x c= +* k dx k x c= +* 11, 1
1
n n x dx x c n
n
+= + ) "
+
*
1lndx x c
x= +
- ./
1 ln x dx x c"= +*
11, 1
1
n n x dx x c n
n
" " +
= + )" +
*
1 1lndx ax b c
ax b a= + +
+
- ./
11
1
p p p q
q q q
p
q
q x dx x c x c
p q
++
= + = +
+ +*
Trig Functions
cos sinu du u c= +* sin cosu du u c= " +* 2
sec tanu du u c= +*
sec tan secu u du u c= +* csc cot cscu udu u c= " +* 2csc cotu du u c= " +*
tan ln
secu du u c= +* cot ln sinu du u c= +*
sec ln sec tanu du u u c= + +* ( )3 1sec sec tan ln sec tan2
u du u u u u c= + + +*
csc ln csc cotu du u u c= " +* ( )3 1csc csc cot ln csc cot
2u du u u u u c= " + "*
Exponential/Logarithm Functions
u udu c= +* e e
ln
uu a
a du ca
= +* ( )ln lnu du u u u c= " +*
( ) ( ) ( )( )2 2sin sin cos
auau bu du a bu b bu c
a b= " +
+*
ee ( )1u uu du u c= " +* e e
( ) ( ) ( )( )2 2cos cos sin
auau bu du a bu b bu c
a b
= + +
+
*e
e 1
ln ln
ln
du u c
u u
= +- .
/
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Common Derivatives and Integrals
Inverse Trig Functions
1
2 2
1sin
udu c
aa u
! " #= +$ %
& '!
( )*
1 1 2sin sin 1u du u u u c! != + ! ++
1
2 2
1 1tan
udu c
a u a a
! " #= +$ %
+ & '
( )*
( )1 1 21tan tan ln 1
2u du u u u c! !
= ! + ++
1
2 2
1 1sec
udu c
a au u a
! " #= +$ %
& '!
( )*
1 1 2cos cos 1u du u u u c! != ! ! ++
Hyperbolic Trig Functions
sinh coshu du u c= ++ cosh sinhu du u c= ++ 2sech tanhu du u c= ++
sech tanh sechu du u c= ! ++ csch coth cschu du u c= ! ++ 2csch cothu du u c= ! ++
( )tanh ln coshu du u c= ++ 1
sech tan sinhu du u c!= ++
iscellaneous
2 2
1 1ln
2
u adu c
a u a u a
+=
! !( )*
2 2
1 1ln
2
u adu c
u a a u a
!= +
! +
( )*
2
2 2 2 2 2 2ln2 2u aa u du a u u a u c+ = + + + + +
+
22 2 2 2 2 2
ln2 2
u au a du u a u u a c! = ! ! + ! ++
22 2 2 2 1sin
2 2
u a ua u du a u c
a
! " #! = ! + +$ %& '
+
22 2 12 2 cos
2 2
u a a a uau u du au u c
a
!! !" #! = ! + +$ %
& '+
Standard Integration Techniques Note that all but the first one of these tend to be taught in a Calculus II class.
u Substitution
Given ( )( ) ( )b
a f g x g x dx,+ then the substitution ( )u g x= will convert this into the
integral, ( )( ) ( ) ( )( )
( )b g b
a g a f g x g x dx f u du, =+ + .
Integration by Parts
The standard formulas for integration by parts are,b bb
aa audv uv vdu udv uv vdu= ! = !+ + + +
Choose u and dv and then compute du by differentiating u and compute v by using the
fact that v dv= + .
Common Derivatives and Integrals
Trig Substitutions
If the integral contains the following root use the given substitution and formula.
2 2 2 2 2sin and cos 1 sina
a b x xb
! ! ! ! - = = !
2 2 2 2 2sec and tan sec 1a
b x a xb
! ! ! ! - = = !
2 2 2 2 2tan and sec 1 tanaa b x xb
! ! ! + - = = +
Partial Fractions
If integrating( )
( )
P xdx
Q x
( )*
where the degree (largest exponent) of ( ) P x is smaller than the
degree of ( )Q x then factor the denominator as completely as possible and find the partial
fraction decomposition of the rational expression. Integrate the partial fractiondecomposition (P.F.D.). For each factor in the denominator we get term(s) in the
decomposition according to the following table.
Factor in ( )Q x Term in P.F.D Factor in ( )Q x Term in P.F.D
ax b+ A
ax b+
( )k
ax b+ ( ) ( )
1 2
2
k
k A A A
ax b ax b ax b+ + +
+ + +
!
2ax bx c+ + 2
Ax B
ax bx c
+
+ +
( )2 k
ax bx c+ + ( )
1 1
2 2
k k
k
A x B A x B
ax bx c ax bx c
+++ +
+ + + +
!
Products and (some) Quotients of Trig Functions
sin cosn m x x dx+
1. If n is odd. Strip one sine out and convert the remaining sines to cosines using2 2sin 1 cos x x= ! , then use the substitution cosu x=
2.
If m is odd. Strip one cosine out and convert the remaining cosines to sines
using
2 2
cos 1 sin x x= !
, then use the substitution sinu x=
3.
If n andm are both odd. Use either 1. or 2.
4. If n andm are both even. Use double angle formula for sine and/or half angleformulas to reduce the integral into a form that can be integrated.
tan secn m x x dx+
1.
If n is odd. Strip one tangent and one secant out and convert the remaining
tangents to secants using 2 2tan sec 1 x x= ! , then use the substitution secu x=
2. If m is even. Strip two secants out and convert the remaining secants to tangents
using 2 2sec 1 tan x x= + , then use the substitution tanu x=
3.
If n is odd and m is even. Use either 1. or 2.4.
If n is even and m is odd. Each integral will be dealt with differently.
Convert Example : ( ) ( )3 3
6 2 2cos cos 1 sin x x x= = !
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! ," ,
!
"
!
"
!
"
!
! "
! "
"
"
x = ρ cos ϕ ρ =√
x2 + y2
y = ρ ϕ ϕ = arctan(y/x)z = z z = z
uρ = cos ϕux + ϕuy
uϕ = − ϕux + cos ϕuy
uz = uz
r = ρuρ + z uz
dr = dlρuρ + dlϕuϕ + dlzuz = dρuρ + ρdϕuϕ + dz uz
dS ρ = dlϕdlz = ρdϕdz ; dS ϕ = dlρdlz = dρdz ; dS z = dlρdlϕ = ρdρdϕ
dτ = dlρdlϕdlz = ρdρdϕdz
sin !
"
"
!
!
cos !
"
!
sin ! "
!
"
!
sin ! "
"
x = r θ cos ϕ r =√ x2 + y2 + z 2
y = r θ ϕ θ = arctan(√ x2 + y2/z )
z = r cos θ ϕ = arctan(y/x)
ur = θ cos ϕux + θ ϕuy + cos θuz
uθ = θ cos ϕux + θ ϕuy − θuz
uϕ = − ϕux + cos ϕuy
r = rur
dr = dlrur + dlθuθ + dlϕuϕ = drur + rdθuθ + r θdϕuϕ
dS r = dlθdϕ = r2 θdθdϕ ; dS θ = dlrdlϕ = r θdrdϕ ; dS ϕ = dlrdlθ = rdrdθ
dτ = dlrdlθdϕ = r2 θdrdθdϕ
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f = f (x,y,z ) A(x,y,z ) = Ax(x,y,z )ux +Ay(x,y,z )uy +Az(x,y,z )uz
∇f = ∂ f
∂ xux +
∂ f
∂ yuy +
∂ f
∂ z uz
∇ · A = ∂ Ax
∂ x + ∂ Ay
∂ y + ∂ Az
∂ z
∇×A =
ö Az
∂ y −
∂ Ay
∂ z
!ux +
ö Ax
∂ z −
∂ Az
∂ x
!uy +
ö Ay
∂ x −
∂ Ax
∂ y
!uz
∇ · (∇f ) ≡ ∇2f =
∂ 2f
∂ x2 +
∂ 2f
∂ y2 +
∂ 2f
∂ z 2
f = f (ρ,ϕ, z ) A(ρ,ϕ, z ) = Aρ(ρ,ϕ, z )uρ+Aϕ(ρ,ϕ, z )uϕ+Az(ρ,ϕ, z )uz
∇f = ∂ f
∂ρuρ +
1
ρ
∂ f
∂ϕuϕ +
∂ f
∂ z uz
∇ · A = 1
ρ
∂ (ρAρ)
∂ρ +
1
ρ
∂ Aϕ
∂ϕ +
∂ Az
∂ z
∇×A =
√1
ρ
∂ Az
∂ϕ −
∂ Aϕ
∂ z
!uρ +
√∂ Aρ
∂ z −
∂ Az
∂ρ
!uϕ
+1
ρ
√∂ (ρAϕ)
∂ρ −
∂ Aρ
∂ϕ
!uz
∇2f =
1
ρ
∂
∂ρ
√ρ∂ f
∂ρ
!+
1
ρ2∂ 2f
∂ϕ2 +
∂ 2f
∂ z 2
f = f (r,θ,ϕ) A(r,θ,ϕ) = Ar(r,θ,ϕ)u
r + Aθ(r,θ,ϕ)uθ +
Aϕ(r,θ,ϕ)uϕ
∇f = ∂ f
∂ rur +
1
r
∂ f
∂θuθ +
1
r θ
∂ f
∂ϕuϕ
∇ · A = 1
r2
∂ (r2Ar)
∂ r +
1
r θ
∂ ( θAθ)
∂θ +
1
r θ
∂ Aϕ
∂ϕ
∇×A = 1
r θ√∂ ( θAϕ)
∂θ
− ∂ Aθ
∂ϕ!ur +
1
r√
1
θ
∂ Ar
∂ϕ
− ∂ (rAϕ)
∂ r!uθ
+1
r
√∂ (rAθ)
∂ r −
∂ Ar
∂θ
!uϕ
∇2f =
1
r2
∂
∂ r
√r2
∂ f
∂ r
!+
1
r2 θ
∂
∂θ
√ θ
∂ f
∂θ
!+
1
r2 2θ
∂ 2f
∂ϕ2
A B C
A · (B ×C) = B · (C ×A) = (A ×B) · C
A × (B ×C) = B(A · C) −C(A · B)
f = f (r) g = g (r) A = A(r) B = B (r)
∇(f + g) = ∇f + ∇g
∇(fg ) = f (∇g) + g(∇f )
∇ · (A + B) = ∇ · A + ∇ · B
∇ · (f A) = f (∇ · A) + A · (∇f )
∇ · (A ×B) = B · (∇× A) − A × (∇× B)
∇× (A + B) = ∇×A +∇×B
∇× (f A) = f (∇× A) − A × (∇f )
∇· (∇× A) = 0
∇× (∇f ) = 0
∇× (∇× A) = ∇(∇ · A) −∇2
A
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Diff erential Equations Study Guide1
First Order Equations
General Form of ODE: dy
dx = f (x,y )(1)
Initial Value Problem: y 0 = f (x,y ), y(x0) = y0(2)
Linear Equations
General Form: y 0 + p(x)y = f (x)(3)
Integrating Factor: µ(x) = eR p(x)dx(4)
=⇒ d
dx (µ(x)y) = µ(x)f (x)(5)
General Solution: y = 1
µ(x)
Z µ(x)f (x)dx + C
(6)
Homeogeneous Equations
General Form: y 0 = f (y/x)(7)
Substitution: y = zx(8)
=⇒ y0 = z + xz 0(9)
The result is always separable in z :
(10) dz
f (z )− z =
dx
x
Bernoulli Equations
General Form: y 0 + p(x)y = q (x)yn(11)
Substitution: z = y1−n(12)
The result is always linear in z :
(13) z 0 + (1− n) p(x)z = (1 − n)q (x)
Exact Equations
General Form: M (x,y )dx + N (x,y )dy = 0(14)
Text for Exactness: ∂ M
∂ y =
∂ N
∂ x(15)
Solution: φ = C where(16)
M = ∂φ∂ x
and N = ∂φ∂ y
(17)
Method for Solving Exact Equations:
1. Let φ =R
M (x,y )dx + h(y)
2. Set ∂φ
∂ y = N (x,y )
3. Simplify and solve for h(y).
4. Substitute the result for h(y) in the expression for φ from step1 and then set φ = 0. This is the solution.
Alternatively:
1. Let φ =R
N (x,y )dy + g(x)
2. Set ∂φ
∂ x = M (x,y )
3. Simplify and solve for g(x).
4. Substitute the result for g (x) in the expression for φ from step1 and then set φ = 0. This is the solution.
Integrating Factors
Case 1: If P (x,y ) depends only on x, where
(18) P (x,y ) = M y −N x
N =⇒ µ(y) = e
R P (x)dx
then
(19) µ(x)M (x,y )dx + µ(x)N (x,y )dy = 0
is exact.
Case 2: If Q(x,y ) depends only on y , where
(20) Q(x,y ) = N x −M y
M =⇒ µ (y) = e
R Q(y)dy
Then
(21) µ(y)M (x,y )dx + µ(y)N (x, y)dy = 0
is exact.
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Second Order Linear Equations
General Form of the Equation
General Form: a(t)y00 + b(t)y0 + c(t)y = g(t)(22)
Homogeneous: a (t)y00 + b(t)y0 + c(t) = 0(23)
Standard Form: y 00 + p(t)y0 + q (t)y = f (t)(24)
The general solution of (22) or (24) is
(25) y = C 1y1(t) + C 2y2(t) + y p(t)
where y1(t) and y2(t) are linearly independent solutions of (23).
Linear Independence and The Wronskian
Two functions f (x) and g (x) are linearly dependent if thereexist numbers a and b, not both zero, such that af (x)+ bg(x) = 0for all x. If no such numbers exist then they are linearly inde-pendent.
If y1 and y2 are two solutions of ( 23) then
Wronskian: W (t) = y1(t)y02(t)− y01(t)y2(t)(26)
Abel’s Formula: W (t) = Ce−R p(t)dt(27)
and the following are all equivalent:
1. {y1,y 2} are linearly independent.
2. {y1,y 2} are a fundamental set of solutions.
3. W (y1,y 2)(t0) 6= 0 at some point t0.
4. W (y1,y 2)(t) 6= 0 for all t.
Initial Value Problem
(28)
y00 + p(t)y0 + q (t)y = 0y(t0) = y0
y0(t0) = y1
Linear Equation: Constant Coefficients
Homogeneous: ay 00 + by0 + cy = 0(29)
Non-homogeneous: ay 00 + by0 + cy = g(t)(30)
Characteristic Equation: ar 2 + br + c = 0(31)
Quadratic Roots: r = −b ±
√ b2 − 4ac
2a(32)
The solution of (29) is given by:
Real Roots(r1 6= r2) : yH = C 1er1t + C 2er2t(33)
Repeated(r1 = r2) : yH = ( C 1 + C 2t)er1t(34)
Complex(r = α ± iβ ) : yH = eαt(C 1 cosβ t + C 2 sinβ t)(35)
The solution of (30) is y = yP + hH where yh is given by (33)through (35) and yP is found by undetermined coefficients orreduction of order.
Heuristics for Undetermined Coefficients(Trial and Error)
If f (t) = t hen g ue ss th at yP =
P n(t) ts(A0 + A1t + · · · + Antn)
P n(t)eat ts(A0 + A1t + · · · + Antn)eat
P n(t)eat sin bt tseat[(A0 + A1t + · · · + Antn) cos bt
or P n(t)eat cos bt +(A0 + A1t + · · · + Antn) sin bt]
Method of Reduction of Order
When solving (23), given y1, then y2 can be found by solving
(36) y1y02 − y01y2 = Ce−R p (t)dt
The solution is given by
(37) y2 = y1
Z e−
R p(x)dxdx
y1(x)2
Method of Variation of Parameters
If y1(t) and y2(t) are a fundamental set of solutions to (23) thena particular solution to (24) is
(38) yP (t) = −y1(t)
Z y2(t)f (t)
W (t) dt + y2(t)
Z y1(t)f (t)
W (t) dt
Cauchy-Euler Equation
ODE: ax2y00 + bxy0 + cy = 0(39)
Auxilliary Equation: ar (r − 1) + br + c = 0(40)
The solutions of (39) depend on the roots of (40):
Real Roots: y = C 1xr1 + C 2xr2
(41)
Repeated Root: y = C 1xr + C 2xr ln x(42)
Complex: y = xα[C 1 cos(β ln x) + C 2 sin(β ln x)](43)
Series Solutions
(44) (x− x0)2y00 + (x− x0) p(x)y0 + q (x)y = 0
If x0 is a regular point of (44) then
(45) y1(t) = (x− x0)n∞
Xk=0
ak(x− xk)k
At a Regular Singular Point x0:
Indicial Equation: r2 + ( p(0)− 1)r + q (0) = 0(46)
First Solution: y1 = (x− x0)r1∞Xk=0
ak(x− xk)k(47)
Where r1 is the larger real root if both roots of ( 46) are real oreither root if the solutions are complex.