diff eq final

how to complete the square

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

partial fraction for denominator Ax+b

A/Ax+b

partial fraction for (Ax+B)^k

A/Ax+B + B/(Ax+B)² + … + K/(Ax+B)^k

partial fraction for ax²+bx+c

Ax+b/ax²+bx+c

partial fraction for (ax²+bx+c)^k

Ax+B/ax²+bx+c + … + Kx+L/(ax²+bx+c)^k

what is the convolution integral

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

laplace formula for heaviside

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

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

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

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

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

what are the equilibrium solutions

the point where the x and y nullclines intersect

what is a nodal source

when both eigenvalues are positive real numbers

what is a nodal sink

when both eigenvalues are negative real numbers

what is a saddle point

when there is one positive eigenvalue and one negative

what is a sprial sink

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

what is a spiral source

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

what is a center point

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

what is the desirable form of a first order nonhomogeneous equation

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

what is the equation for a fourier series

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

equation for Aₙ

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

equation for Bₙ

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

equation for A₀

1/L ∫f(x)dx evaluated from -L to L

define sin or cos as even or odd function

sin is odd, cos is even

what does cos(nπ) reduce to

(-1)ⁿ

what does sin(nπ) reduce to

0

what does an odd integral evaluate to

0

what does an even integral evaluate to

2/L ∫even function evaluated from 0 to L

how do you evaluate a fourier series for a piecewise

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

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

what is the Jacobian

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

what is neimann’s conditions in the heat equation

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

what is the final heat equation

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

how to find k from heat equation IVP

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

how to find T₀ and Tₙ from the hear equation IVP

u(0,t)=T₀ and u(L,t)=Tₙ

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

u(x,0)=f(x)

what is the heat equation final solution

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

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

C₁e^(r₁t) + C₂e^(r₂t)

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

C₁e^(r₁t) + tC₂e^(r₂t)

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

e^(αt) [C₁cos(βt)+C₂sin(βt)]

how to solve for the spring constant

k=(mg)/x

what is the spring equation

y’’+(µ/m)y’+(k/m)y=0 or F(x)