Vu Lam's math map

First order DE of the form

dy/dt = f(y,t) = slope

General solution: y = Ce^2t

Isoclines "equal slopes"

Place where slope is the same

Equilibrium

occur at y = k where dy/dt = 0solution have to remain all the time

Concative change

2nd derivative = 0

Stability

Concave upConcave down.

Separable equation

dy/dt = f(t)g(y)

set g(y) = 0 and solve for equalibrium solution, if any

Now assume g(y) not equal 0, rewrite the equation in saparated or differential form: dy/g(y) = f(t)dt

Intergrate each side

if possible,solve for y in term of t

if having IVP, find the value

Intergrating directly

dy/dt = f(t)

intergrate both side

If having IVP, find the value

Intergrating factor

dy/dt + p(t)y = f(t)

u(t)=e^interal(p(t)dt

multiply both side by u(t)

intergrate both side respect to t

if having IVP, find the value

Using Laplace transforms

L[f']=sL[f] - f(0) =sF(s) - f(0)

Growth and decay

dy/dt = Ky

K>0 , k is growth factor

K<0, k is decay factor

y(to) = yo

y=yoe^kt

dA/dt = rA+d

A(0) = Ao

r is interest rate compound continously

d is the additional dollars per year from the depositor

A(t) = Aoe^rt + d/r(e^rt - 1)

mixing and cooling

dx/dt = rate in - rate out

rate in = (concentrate In) * (flow rate in)

rate out = (concentrate out) * (flow rate out)

Newton's law of cooling

dT/dt = k(M-T)

K is the constant of propotionality

M is the medium temp

temperature of the object

T(t) = Toe^-kt + M(1-e^-kt)

System of differential equations

dx/dt = P(x,y)
dy/dt = Q(x,y)

autonomous: no t involve

first order

system in 2 variables

set dx/dt = 0, solve for y: V null

set dy/dt = 0,solve for y : H null

graph those. choose vnull point plug into hnull. if it <0, down arrow. choose Hnull point plug into vnull. if it < 0, left arrow.

Second order DE

ay'' + by' +cy = 0

use characteristic equation: ar^2 + br + c = 0

solve for discriminante : delta = b^2 - 4ac

if delta > 0: y(t) = c1e^r1t + c2e^r2t

if delta=0: y(t) = c1e^rt + c2te^rt

if delta<0: y(t) = e^at(c1cos(bt)+c2sin(bt))

Simple harmonic oscillator: mx'' + bx' + kx =f(t)

Undertermined coefficent

electric circuits: LQ'' + RQ' + 1/CQ =V(t)

ay'' +by' +cy = f(t)

y = yh +yp

yh =homogenous

using characteristic equation to sove for

f(t) is poly family

yp = At^2 + Bt + C
y'p = 2At + B
y''p =2A
solve for A, B, C

f(t) is exp family

yp = Ae^t
y'p=Ae^t
y''p=Ae^t
solve for A

if serpent bite: yp = Ate^tif bite twice: yp = At^2e^t

f(t) is trig family

yp = Asint+Bcost
y'p=Acost - Bsint
y''p=-Asint-Bcost
solve for A and B

Using laplace transforms

L[f^n] = s^n L[f] - s^(n-1) f(0) - s^(n-2)f'(0) ... - sf^(n-2)(0)-f^(n-1)(0)

Laplace transforms

F(s) = interal e^-st f(t) dt from 0 to infinity

use the laplace table

Unit step function

draw the pic from given information

write step function for it

solve for y(t) by using the table

Matrices

A^T : transpose of A: interchange the row and column vector in an m x n matrix Asquare matrix: same member of m x nmain diagros { 1 0 0 0 1 0 0 0 1}tr(A): trace of A: sum main diaro

elementary row operation

ERO1) trade rows places (permit): change row2) multiply every element in row by no zero scalar3) add element of 1 row to correspinding element of another row

RREF

1) its a row echelon matrix2) any column that contain a leading 1 has 0 everywher else

pivot column: column contain leading 1

dependent variable: bound variable

Independent variable: free variable

consistance: have solution or infinite solution

inconsistance: no solution

Inverse of matrix A = A^-1

x = A^-1*b

[A|I] = [I|A^-1]

A^-1 = 1/det [W - y
-Z x]

only work for 2 x 2 matrix

Invertible: detA not equal 0
not Invertible: detA = 0

Subspace

let u in W where u = (u1,u2,u3)
let v in w where v = (v1,v2,v3)
let Cin R

show u+v in W

show CU in W

if those are equal, hence closed with respect to scalar multiplication W in a subspace of V

Linear transformation

let C in Rlet u =(u1,u2) in Vlet v =(v1,v2) in V1) show cT(u) = T(cu)2)show T(u+v) = T(u) +T(v)if they are the same -> linear transformation

colA

REF of A

according to leading 1 of REF, refer to the original matrix a. those vector are the bases of col(A)if it = codomain, surjective

rank

n-the nunber of non pivot colum if Athe nbuber of pivot column in Adim(im(T)) = dim(colA)

Ker(T)

Ref of A

write as system equation and parameter equation.

Nullity

how many vector is in ker(T)if it = 0, injective

eigen value & eigen vector

T(v) = lamda(v)

v =eigen vetorlamda = eigen value

find eigen value

|A-lamda*I|=0

find eigen vector

plug lamda back to |A-lamda*I|=0

general solution

x = C1e^lamda1tv1 +C2e^lamda2tv2

repeated lamda

x = C1e^lamdatv1 +C2te^lamdatv2

not enough lamda

find u

x = C1e^lamdatv +C2te^lamdatv + C2e^lamdatu