MA 266 E2

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Center

purely imaginary

Spiral sink

complex with negative real part

Spiral source

complex with positive real part

Node sink

two negative real eigenvalues

Saddle

real eigenvalues with opposite signs

mx`` + cx` + kx = F(t)

m = mass
c = damping coefficient/dashpot
k = spring constant
F(t) = outside force
x = position

vocab for mx`` + cx` + kx = F(t) (and what is resonance)

undamped: c = 0
Damped: C > 0
Free/unforced: F(t) = 0
Forced : F(t) ≠ 0

resonance is when the natural frequency and the forcing frequency are the same.

Classify a damped mass-spring system using c^2 - 4mk

c^2 - 4mk > 0: overdamped

c^2 - 4mk = 0: critically damped

c^2 - 4mk < 0: underdamped

converting x_sp(t) = Acos(wt) + Bsin(wt)
to
Ccos(wt - a)

C = sqrt(A² + B²)

a = arctan(B/A)

x_sp(t) is the steady periodic solution. Found by using undetermined coefficients on the right side and getting values for A and B with the diff eq.

classify origin of x` = Ax using its eigenvalues

λ = ±bi —> center
λ = a ±bi, a > 0 —> spiral source

λ = a ±bi, a < 0 —> spiral sink

Two real λ of opposite signs —> saddle

Both λ are negative and real —> node sink

Both λ are positive and real —> node source

defective λ—> means?

repeated λ but only one linearly dependent eigenvector
get v normally: (A - λI)v = 0
get w with v: (A - λi)w = v

general form: x(t) = C₁(e^λt)v + C₂(e^λt)(tv + w)

Variation of Parameters (form) : y'' + P(x)y' + Q(x)y = g(x)

1. find y1 and y2, which are the things attached to c1 c2 in the homogenous solution.
2. Find W, which is y1y2` - y2y1`
3. u1' = -(y2*g)/W
4. u2' = (y1*g)/W
5. yp = u1*y1 + u2*y2

g = the right side in the form

If full solution is requested:
y = C1*y1 + C2*y2 + yp

e^At for triangular

see image

e^at for general case

identify eigenvalues based on phase portrait

Closed loops: center

No spiraling; everything moves toward the origin: node sink

No spiraling; everything moves away from the origin: node source

One straight direction moves toward the origin while 
another straight direction moves away: saddle

Arrows shrink and reverse across an entire line instead
 of converging to one point: one eigenvalue is zero

Arrows toward that line: 
other eigenvalue is negative

Arrows away from that line: 
other eigenvalue is positive

For direction-field matching, plug in (1,0) or (0,1) and match the arrow direction.

trajectory Identification

x^2 + y^2 = C
Circle

A*x^2 + B*y^2 = C, where A and B are positive and unequal
Ellipse

A*x^2 - B*y^2 = C
Hyperbola

y = m*x
Straight line through the origin

y^2 = A*x or x^2 = A*y
Parabola