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1

how to complete the square

take half of the coefficient of s, square it, then add and subtract the result to the polynomial

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2

partial fraction for denominator Ax+b

A/Ax+b

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3

partial fraction for (Ax+B)^k

A/Ax+B + B/(Ax+B)² + … + K/(Ax+B)^k

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4

partial fraction for ax²+bx+c

Ax+b/ax²+bx+c

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5

partial fraction for (ax²+bx+c)^k

Ax+B/ax²+bx+c + … + Kx+L/(ax²+bx+c)^k

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6

what is the convolution integral

(f*g)(t)=∫f(t-Γ)g(Γ)dΓ evaluated from 0 to t

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7

laplace formula for heaviside

L[f(t-c)H(t-c)] = e^(-cs)F(s)

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8

how to turn a piecewise into a heaviside

write the function outside of the heaviside, in brackets write the H function as [H(t-x₀)-H(t-x₁)] from the interval the function occurs

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9

what is the inverse laplace of the heaviside function [e^(-cs)F(s)]

f(t-c)H(t-c)

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10

when building a convolution from a delta function, what do you do with the function on the right side

multiply the convolution by the delta function

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11

what are the equilibrium solutions

the point where the x and y nullclines intersect

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12

what is a nodal source

when both eigenvalues are positive real numbers

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13

what is a nodal sink

when both eigenvalues are negative real numbers

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14

what is a saddle point

when there is one positive eigenvalue and one negative

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15

what is a sprial sink

when the two complex eigenvalues have a negative real value (α<0)

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16

what is a spiral source

when the two complex eigenvalues have a positive real value (α>0)

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17

what is a center point

when the two complex eigenvalues have no real value (α=0)

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18

what is the desirable form of a first order nonhomogeneous equation

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

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19

what is the equation for a fourier series

f(x) = A₀/2 + ∑Aₙcos(nπx/L) + ∑Bₙsin(nπx/L)

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20

equation for Aₙ

1/L ∫f(x)cos(nπx/L)dx evaluated from -L to L

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21

equation for Bₙ

1/L ∫f(x)sin(nπx/L)dx evaluated from -L to L

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22

equation for A₀

1/L ∫f(x)dx evaluated from -L to L

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23

define sin or cos as even or odd function

sin is odd, cos is even

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24

what does cos(nπ) reduce to

(-1)ⁿ

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25

what does sin(nπ) reduce to

0

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26

what does an odd integral evaluate to

0

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27

what does an even integral evaluate to

2/L ∫even function evaluated from 0 to L

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28

how do you evaluate a fourier series for a piecewise

split the integrals into 2 with limits from -L→0 and 0→L

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29

steps for solving a nonlinear system

solve for equilibrium points, solve the Jacobian, solve Jacobian at equilibrium points, find eigenvalues from that matrix and categorize those points

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30

what is the Jacobian

matrix form of the partial derivatives of x and y for both equations

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31

what is neimann’s conditions in the heat equation

when the rod is insulated so the partial of u/x is equal to 0

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32

what is the final heat equation

the partial of u/t = k*second partial of u/x + u(x,t)/cp

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33

how to find k from heat equation IVP

u(x,t) = k*second partial of u/x

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34

how to find T₀ and Tₙ from the hear equation IVP

u(0,t)=T₀ and u(L,t)=Tₙ

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35

how to find the function f(x) from the heat equation IVP

u(x,0)=f(x)

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36

what is the heat equation final solution

u(x,t)= fourier series of f(x) *e^kn²t

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37

what is the homogeneous solution for a second order DE with 2 real roots

C₁e^(r₁t) + C₂e^(r₂t)

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38

what is the homogeneous solution for a second order DE with 1 real root

C₁e^(r₁t) + tC₂e^(r₂t)

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39

what is the homogeneous solution for a second order DE with 2 complex roots

e^(αt) [C₁cos(βt)+C₂sin(βt)]

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40

how to solve for the spring constant

k=(mg)/x

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41

what is the spring equation

y’’+(µ/m)y’+(k/m)y=0 or F(x)

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