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| Linear independence between two functions |
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Definition
c*f(x) != g(x) is independent
Examples:
(t,3t) are linearly DEPENDENT
(t, t^2) are linearly INDEPENDENT |
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Definition
* Where unique solutions are guaranteed.
Steps:
1) Put equation into standard form:
y'' term ON IT'S OWN -> NO OTHER FUNCTIONS
y'' + p(t)y' + q(t)y = g(t)
2) Use "holes" in standard eqn and the given numbers (ex: y(0) = 1) to find where it MUST exist. |
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Definition
[image]
theta is the angle from the (0,1)
example:
1+i
a = 1 b = 1
sqrt(2) * (cos(pi/4)+isin(pi/4)) |
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| Eulers formula for complex |
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Definition
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| Characteristic equation has two 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 a polynomial ONLY. What is Y? |
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Definition
Y is going to be a polynomial of degree n.
Example:
g(t) = t^3 + 2t + 1
Y = At^3 + Bt^2 + Ct + D |
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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 * sin(bt). What is Y? |
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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 * cos(bt). What is Y? |
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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) * 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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| What happens if we find Y and it has a potential duplicate with y_c |
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Definition
Keep mulitplying Y by t until you don't!
ex: e^t and Ce^t
becomes te^t and Ce^t |
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Give the meanings of everything about an undamped vibration
* General eqn, general solution |
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Definition
mu'' + ku = 0
Acos(w0t) + Bsin(w0t)
Rcos(w0t - d) |
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| w0, period of oscillation, |
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Definition
w0 = sqrt[k/m]
T = 2pi * sqrt[k/m] |
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Definition
| Lq'' + Rq' + 1/c Q = E(t) |
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Definition
| Say your vibrations are being forced @ Fcos(wt). If w = w0, then it's resonant. |
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Transient solution
Steady state solution |
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Definition
Say you solved a diff eq for u_c and U, where u_c is the complimentary sol'n from the left side of diff eq, and U is the right side.
u_c is the transient solution.
U is the steady state solution. |
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| Summarize finding solution to DE using the reduction of order method (given a sol'n) |
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Definition
1) Use equation on sheet for it.
5) Find solution for DE in terms of w.
6) Integrate your solution to find v
7) Find your solution. It will be in the form vy_1 + y_1 |
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| Euler EQN. Identify and how to start solving. |
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Definition
The euler equation is: [image]
To find y1 and y2 from this:
a*s*(s - 1) + b*s + c = 0
a,b,c, are constants from DE
Solve for s. Solutions will be y = t^s
If variation of parameters for next step, DIVIDE by t^2 to get it!!!! |
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