Tablas Resumen Electromagnetismo I

7/23/2019 Tablas Resumen Electromagnetismo I

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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

" " ! ! ! !

" " ! ! ! !

" " ! ! ! !

' ( ' (! = ! =) * ) *

+ , + ,

' ( ' (! = ! =) * ) *

+ , + ,

' ( ' (! = ! =) * ) *

+ , + ,



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

#

$

%



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

= "


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

= +- .

/




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= + .


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= = !



! ," ,

!

"

!

"

!

"

!

! "

! "

"

"

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



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 =

√∂ Az

∂ y −

∂ Ay

∂ z

!ux +

√∂ Ax

∂ z −

∂ Az

∂ x

!uy +

√∂ 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



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.

Tablas Resumen Electromagnetismo I

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