Differential Equations Exam Study Guide

Flashcards that contain key terms and definitions from lecture notes. Terms are left blank for fill-in-the-blank style practice.

P(x) and f(x) continuous on open interval

linear, first order existence & uniqueness theorem

f(x, Y) and ∂f/∂x continuous near x0

nonlinear, first order existence & uniqueness theorem

all coefficients an and g(x) continuous on interval containing x0, and all an not equal to 0

higher order DE existence and uniqueness theorem

y = _ auxiliary equation answer for 2 distinct real roots

Ae^m1x + Be^m2x

y = _ auxiliary equation answer for 2 complex roots a+iB

e^ax( c1cos(Bx) + c2sin(Bx) )

_ auxiliary equation answer for 2 repeated real roots

(c1 + c2x)*e^mx

if W does not equal 0 for some x0 in I, the functions are _

Wronskian definition

_ laplace transform formula

∫ (e^-st * f(t))dt from 0,infinity

even periodic extension

cosine series is considered

odd periodic extension

sine series is considered

f(-x) = -f(x)

odd function

f(-x) = f(x)

even function

reflect over y axis

even periodic extension steps

reflect over x axis

odd periodic extension steps

U(t-a) = 0 for t<a 1 for t ≥ a

unit step function is

s.t. A*Xp + F(t) = Xp'

formula to solve for Xp variables in system of equations

e^(a-i)t =

e^at * (cos(t) - i*sin(t))

X1 = (Xa+Xb)/2 X2 = (Xa-Xb)/2i then: X = c1X1 + c2X2

formula for X1 and X2 when eigenvalues are fake af

c1e^λ1t[X1] + c2e^λ2t[X2]

formula for X in systems of equations (normal)

if g(x) = 3, we guess

yp = A

if g(x) = 5x + 9, we guess

yp = 5x + 9

if g(x) = x^3 + 2x + 1, we guess

yp = Ax^3 + Bx^2 + Cx + D

if g(x) = cos(3x), we guess

yp = Acos(3x) + Bsin(3x)

if g(x) = e^5x, we guess

yp = Ae^5x

if g(x) = (3x^2 - 2)e^x, we guess

yp = (Ax^2 + Bx + C)e^x

if g(x) = cos(4x)xe^3x, we guess

yp = cos(4x)(Ax+B)e^3x + sin(4x)(Cx+D)e^3x