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| Integrating factors (steps/method) |
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1) Get into standard form y' + p(t)y = q(t)
2) Use integrating factor [image]
3) Multiply both sides by integrating factor.
4) Use "reverse chain rule" to get it into something like (y*mu(t))' = g(t)
5) Integrate and solve for y. |
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| Autonomous Equations and how to solve. |
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[image]
No dependence on d, can simply integrate for solutions. Typically, you would use partial fractions here. |
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Use the equation sheet to see what v will be.
Then, solve this linear ODE:
[image]
Finally, when you get v (with a +C), solve for the value of y using your original transformation. |
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| Follow formula. y will depend on something like t or x. |
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| Logistic Differential Equations and Modeling |
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| Usually autonomous DEs, so you can use the phase line. |
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[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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| Characteristic equation has two real roots. Find general sol'n |
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[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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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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| 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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| 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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| When should I use variation of parameters? How would I use it? |
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Definition
Use it when there is something "weird" on the left hand side of the DE (not sin cos poly x)
To use:
1) Solve characteristic on right. Get y1 and y2
2) Find u1' and u2' from the equation on formula sheet.
3) Find u1 and u2. **Remember to include a constant of integration here**
4) Obtain a solution for y, as seen. |
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Make P(x)y'' + Q(X)y' + R(x)y + lambda*W(X)*y = 0
into sturn liouville form |
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Definition
[image]
Multiply the expression by mu(x) |
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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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| Prove two functions are orthogonal |
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Definition
[image]
True if orthogonal |
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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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| Extend odd and even and the connection to fourier series |
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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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