ODE test 2 Flashcards
linear second order ODE
y’’ =py’+qy=gt
p, g, q are all functions of t or constatns
what is a homogeneous 2ed order ode
gt = 0
linear combinaiton
2 solutions to an ode that are added together are still a solution
how to find the range of a variable that satisfies the ode
plug in the derivatives of the equation and solve for the final values of the equation to find the mulah
how to tell if 2 equations are linearly independent
if the wronskian is not equal to 0 then they are linearly independent.
formula for the wronskian
u v
u’ v’
formula for spring metal
my’‘+muy’+ky=fk
what does each part of the spring formula mean?
m: mass
mu: dampening constant
k; spring constant
fk; forcing function
what are the 3 cases for unforced harmonic motion
underdamped <0
overdamped >0
critically damped =0
eulers formula
e^ix = cosx+isinx
what is the formula for complex solutions
e^(at)(c1cos(bt)+c2sin(bt))
guesses for 3t^3
at^3 +bt^2+ct+d
guess for e^2t
ae^(2t)
guess for cos(3t)
acos(3t) + bsin(3t)
equation for the forced undamped harmonic motion
y(t) = y’’ + wo^2y=Acos(wt)
what is wo,, A, and w in the forced undamped harmonic motion equation
wo is the natural frequency
w is the forced frequency
A = amplitude
general solution for undamped forced harmonic motion
c1cos(wot)+c2sin(wot)
2 cases for forced undamped harmonic
w!=wo
or w=w0
w=w0 particular
(At/2wo)sin(wot)
w!=w0 particular
(A/(wo^2-w^2))(cos(wt)-cos(wot))
mean frequency
wbar= (w0+w)/2
half distance
gama=(w0-w)/2
what part is the transient and which is the steady state
the transient is the part where the limit is equal to 0
the steady state is the part that does not change.
k =
mg/l
w0^2 =
sqrt(k/m)
general form for harmonic motion with no dampening constant
c1cos(w0t)+c2sin(w0t)
