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MATH 285 Final Exam Review
MATH285 UIUC Spring 2018
28
Mathematics
Undergraduate 2
05/08/2018

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Cards

Term
Solution to y' = ay + b
Definition
[image]
Term
Integrating factors (steps/method)
Definition
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.
Term
Separation of Variables
Definition
[image]

Cross multiply
Term
Autonomous Equations and how to solve.
Definition
[image]

No dependence on d, can simply integrate for solutions. Typically, you would use partial fractions here.
Term
Bernoulli - how to solve
Definition
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.
Term
Use reduction of order.
Definition
Follow formula. y will depend on something like t or x.
Term
Logistic Differential Equations and Modeling
Definition
Usually autonomous DEs, so you can use the phase line.
Term
Given ai+b find z
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))
Term
Eulers formula for complex
Definition
[image]
Term
Characteristic equation has two real roots. Find general sol'n
Definition
[image]

where r1 and r2 are the roots of the eqn
Term
Characteristic equation has two complex roots. Find general sol'n
Definition
Given two complex roots:

[image]

where

[image] and [image]

The general solution is:

[image]
Term
Characteristic equation has repeated roots. Find general sol'n
Definition
[image]
Term
Say there is a differential equation with L[y] = g(t). g(t) is a polynomial ONLY. What is Y?
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
Term
Say there is a differential equation with L[y] = g(t). g(t) is e^at * P(t). What is Y?
Definition
Y = e^at(P(t))

Example:

e^2t*3t^2

Y = e^2t * (At^2 + Bt + C)
Term
Say there is a differential equation with L[y] = g(t). g(t) is e^at * sin(bt). What is Y?
Definition
[image]
Term
Say there is a differential equation with L[y] = g(t). g(t) is e^at * P(t) * cos(bt). What is Y?
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)
Term
When should I use variation of parameters? How would I use it?
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.
Term
Make P(x)y'' + Q(X)y' + R(x)y + lambda*W(X)*y = 0

into sturn liouville form
Definition
[image]

Multiply the expression by mu(x)
Term
period T
Definition
[image]
Term
How to see if BVP is homogeneous?
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
Term
Prove two functions are orthogonal
Definition
[image]

True if orthogonal
Term
Dirichlet BC
Definition
y(a) = y(b) = 0
Term
Neumann BC
Definition
y'(a) = y'(b) = 0
Term
Mixed BC
Definition
y(a) = y'(b) = 0
Term
Periodic BC
Definition
y(a) = y(b)
y'(a) = y'(b)
Term
If the function f is odd, what can we say about the Fourier coeffs?
Definition
a_n = 0
b_n follows formula

for all n ≥ 0
Term
If the function f is even, what can we say about the Fourier coeffs?
Definition
a_n follows formula
b_n = 0

n ≥ 1
Term
Extend odd and even and the connection to fourier series
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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