ODE test 2

Question 1

linear second order ODE

Answer 1

y’’ =py’+qy=gt

p, g, q are all functions of t or constatns

Question 2

what is a homogeneous 2ed order ode

Answer 2

gt = 0

Question 3

linear combinaiton

Answer 3

2 solutions to an ode that are added together are still a solution

Question 4

how to find the range of a variable that satisfies the ode

Answer 4

plug in the derivatives of the equation and solve for the final values of the equation to find the mulah

Question 5

how to tell if 2 equations are linearly independent

Answer 5

if the wronskian is not equal to 0 then they are linearly independent.

Question 6

formula for the wronskian

Answer 6

u v

u’ v’

Question 7

formula for spring metal

Answer 7

my’‘+muy’+ky=fk

Question 8

what does each part of the spring formula mean?

Answer 8

m: mass
mu: dampening constant
k; spring constant
fk; forcing function

Question 9

what are the 3 cases for unforced harmonic motion

Answer 9

underdamped <0
overdamped >0
critically damped =0

Question 10

eulers formula

Answer 10

e^ix = cosx+isinx

Question 11

what is the formula for complex solutions

Answer 11

e^(at)(c1cos(bt)+c2sin(bt))

Question 12

guesses for 3t^3

Answer 12

at^3 +bt^2+ct+d

Question 13

guess for e^2t

Answer 13

ae^(2t)

Question 14

guess for cos(3t)

Answer 14

acos(3t) + bsin(3t)

Question 15

equation for the forced undamped harmonic motion

Answer 15

y(t) = y’’ + wo^2y=Acos(wt)

Question 16

what is wo,, A, and w in the forced undamped harmonic motion equation

Answer 16

wo is the natural frequency
w is the forced frequency
A = amplitude

Question 17

general solution for undamped forced harmonic motion

Answer 17

c1cos(wot)+c2sin(wot)

Question 18

2 cases for forced undamped harmonic

Answer 18

w!=wo

or w=w0

Question 19

w=w0 particular

Answer 19

(At/2wo)sin(wot)

Question 20

w!=w0 particular

Answer 20

(A/(wo^2-w^2))(cos(wt)-cos(wot))

Question 21

mean frequency

Answer 21

wbar= (w0+w)/2

Question 22

half distance

Answer 22

gama=(w0-w)/2

Question 23

what part is the transient and which is the steady state

Answer 23

the transient is the part where the limit is equal to 0

the steady state is the part that does not change.

Question 24

k =

Answer 24

mg/l

Question 25

w0^2 =

Answer 25

sqrt(k/m)

Question 26

general form for harmonic motion with no dampening constant

Answer 26

c1cos(w0t)+c2sin(w0t)