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| Give an example of linear dependence and independence (rough idea) |
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Definition
Independence - can add up terms without them interfering
Example (independence):
f(x) = x g(x) = x^2 h(x) = x^3
Examples (dependence)
f(x) = sin^2(x) g(x) = cos^2(x) h(x) = 1
f(x) = t^2 g(x) = 3t^2 h(x) = t^3 |
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| Characteristic equation has two (or any) real roots. Find general sol'n |
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Definition
[image]
where r1 and r2 are the roots of the eqn |
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| Characteristic equation has two complex roots. Find general sol'n |
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Definition
Given two complex roots:
[image]
where
[image] and [image]
The general solution is:
[image] |
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| Characteristic equation has repeated roots. Find general sol'n |
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Definition
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| Say there is a differential equation with L[y] = g(t). g(t) is e^at * P(t). What is Y? |
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Definition
Y = e^at(P(t))
Example:
e^2t*3t^2
Y = e^2t * (At^2 + Bt + C) |
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| Say there is a differential equation with L[y] = g(t). g(t) is e^at * P(t) * cos(bt). What is Y? |
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Definition
Y = e^at R(t) sin(bt) + e^at Q(t) cos(bt)
Example:
g(t) = e^2t*cos(3t)*6t
Y = e^2t (At + B) sin(6t) + e^2t (Ct + D) cos(6t) |
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| Prove two functions are orthogonal |
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Definition
[image]
True if orthogonal |
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| How to see if BVP is homogeneous? |
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Definition
Right side of DE is 0 (L[y] = 0)
Boundary values are 0 (y' = 0, y = 0, y + y', etc) on the right side |
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| Theorem for BVP and solutions |
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Definition
If homogeneous BVP has trivial sol'n -> nonhomogeneous has a nontrivial sol'n
If homogeneous BVP has a nontrivial solution -> nonhomogeneous has either infinite or no solutions. |
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BVP vs IVP
*what each stands for
*what about initial conditions
*what about solutions |
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Definition
BVP - Boundary Value Problem
IVP - Initial Value Problem
BVP - Two points are specified.
IVP - One point is specified.
Example:
y(0) = 0 y'(0) = 7 is IVP
y(0) = 0 y'(pi) = 4 is BVP
Solutions:
IVP -> unique solution
BVP -> no solution (constants conflict), one solution or infinite solutions (constant remaining) |
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Definition
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Definition
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Definition
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Definition
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if m and n are integers, what is:
[image]
[image]
[image] |
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Definition
1) [image] is 0
2) [image] is 0, except is L when n = m
3) [image] is 0, except is L when n = m |
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Definition
Even
f(-x) = f(x) - graph symmetric about y axis (ex: x^2)
Odd:
f(-x) = -f(x)
Graph symmetric around origin
ex: x^3 |
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| If the function f is odd, what can we say about the Fourier coeffs? |
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Definition
a_n = 0
b_n follows formula
for all n ≥ 0 |
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| If the function f is even, what can we say about the Fourier coeffs? |
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Definition
a_n follows formula
b_n = 0
n ≥ 1 |
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Definition
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Definition
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| General Principle of convergence |
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Definition
The “smoother” the function is, the faster its Fourier coefficients will decay and the more rapidly the partial sums will approximate the function (fewer terms needed to well approximate the function).
smooth functions converge faster |
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| Values taken by fourier series for f(x) are... |
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Definition
[image] used to describe.
Takes value of function at non-jump points.
Takes value of the midpoint at the jumps.
The actual value is irrelevant.
ex:
{ x^2 @ x > 0
f(x) = { 100 @ x < 0
{ 500 @ x = 0
The fourier series will be y = 50 at x = 0 (100-0/2) |
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| Extend odd and even and the connection to fourier series |
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Definition
Make it so it's an even or odd function
Example
[image]
(use midpt method for odd)
If extended even -> can make fourier cosine series
If extended odd -> can make fourier sine series |
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| Heat Equation and different types of conditions |
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Definition
[image]
Boundary (t can vary)
u(0,t) = 0
u(L,t) = 0
Initial condition:
u(x,0) = f(x)
condition before time changes. |
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Definition
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| Steady state for heat eqn, and other info about them |
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Definition
v(x) = c1 + c2*x
They can also be non-existent or impossible sometimes! |
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| How do mixed BC make the heat equation different than dirirchlet or neumann BC |
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Definition
Take Dirichlet BC
Replace every "L" NOT inside an integral/sum with 2L, and replace n with 2n-1
See:
[image][image] |
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| a in wave eqn - definition |
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Definition
[image]
T is tension
rho is the linear density (mass per unit length)
sqrt(a) is also velocity |
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Definition
| Wave fixed at certain points, sin function oscillates. |
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Definition
nth natural frequency:
n*pi*a/L |
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| What happens when a traveling wave hits a fixed endpoint (Dirichlet boundary condition)? |
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Definition
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Definition
1) Make sure v=0:
Split up wave into 2 parts, with half the height.
Middle split? There will be a plateau |
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| If v = 0 what should I do for wave eqn |
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Definition
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| Draw something for a fixed x |
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Definition
Use both wave equations to show what will be happening. Translate them and move them!
Usually falls off to x = 0 at some time t once the wave has passed |
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Definition
1) Use X(x)*T(t)
2) Plug in X(x) and T(t) into the PDE
3) Get all X terms on one side, and T terms on other. Set this ratio equal to -lambda
4) Find eigenvalues and eigenfunctions (typically for X(x)).
5) Plug those in, and solve the other side.
6) Get the final solution as Xn(x) * Tn(t) will all the eigenvalues/functions |
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