test 3

Question 1

how to with find the gen sol to a homo geneous high order DE with real distinct Eigenvalues.

Answer 1

Guess sol y = e^λx then take the derivative as many times as you need. Then sub into the entire DE solving for λ and those values are you’re eigenvalues. plug then into the fundamental sol: y=(c_1e(^λ_1 * x)) + (c_2e(^λ_2 * x))

Question 2

how to with find the gen sol to a homo geneous high order DE with real repeated Eigenvalues.

Answer 2

Guess sol y = e^λx then take the derivative as many times as you need. Then sub into the entire DE solving for λ and those values are you’re eigenvalues. plug then into the fundamental sol: y=(c_1e(^λ_1 * x)) + (c_2e(^λ_2 * x)). Then multiply the second part by x or the independent variable.

Question 3

how to with find the gen sol to a homo geneous high order DE with complex Eigenvalues.

Answer 3

Guess sol y = e^λx then take the derivative as many times as you need. Then sub into the entire DE solving for λ and those values are you’re eigenvalues. You should have a real part (p) and an imaginary part (q) and should resemble λ = p + or - qi. Plug those into the general solution: y= e^px (c_1cos(qx) + c_2sin(qx))

Question 4

how do you check if a if a set of solutions forms a fundamental solution set?

Answer 4

1. check if number of solns matches the order of DE. (2 solns for 2nd order DE)
2. actually check if they are solns. (LHS = RHS)
3. Check Wronskian doesn’t = 0

Fundamental Solution Set Thrm only works for HOMOGENEOUS EQNS

Question 5

what do you guess the soln as when you solve a DE using a power series?

Answer 5

∞
y = Σ(a_n)*x^n
n=0

Question 6

what do you guess the soln as when you solve a cauchy euler eqn?

Answer 6

y=t^r

Question 7

what should you get as your soln when you solve a DE with complex eigenvalues and with constant coeffs?

Answer 7

λ = p ± qi

p is the real part
q is the complex part

y=e^px*(cos(qx)+sin(qx))

Question 8

what should you guess your soln as when you solve a higher order DE with constant coeffs

Answer 8

y = e^λx

Question 9

what if you have real repeated solns when you solve a higher order DE with constant coeffs?

Answer 9

add an x to your c_2, x^2 to you’re c_3 and so on.

Question 10

when using method of undetermined coeffs, what should you be guessing?

Answer 10

when g(x) = Ae^(kx), guess y_p = Ae^(kx)
when g(x) = Acos(rx) + Bsin(rx), guess y_p = Acos(rx) + Bsin(rx
when g(x) = an nth degree polynomial, guess the most general nth degree polynomial

Question 11

What is Part Stability Thrm for 2D Linear Systems?

Answer 11

consider a 2 x 2 linear system of DE’s

Stable when both e-values are < 0
Unstable when both e-values are > 0
Semi when λ_1 < 0 < λ_2