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26 terms

Test 1

STUDY
PLAY
Equilibrium Solution
y'=0
Stable Equilibrium
Solutions converge to it
Unstable Equilibrium
Solutions diverge from it
Semistable Equilibrium
Solutions converge to it on one side, diverge from it on the other side.
Source (repeller)
Unstable equilibrium on phase line
Sink (attractor)
Stable equilibrium on phase line
Node
Semistable equilibrium on phase line
Needs Euler method
n, h, starting point, y'
Times in Euler's method
t0= starting t
t1=t0+h etc.
y in Euler's method
y0= strting y
y1=y0+h*y'(at starting point)
etc.
Runge Kutta 2nd order
k1= same slope as in Euler
k2= slope at point
(t0+(h/2),y value on first tangent line at half step)
y1=y0+k2*h
Order of DE
Highest derivative
Linear DE
y,y',y'', etc all on first power. (t can be not linear)
Homogeneous DE
The term without any y is 0.
Constant coefficient DE
In front of the ys we have numbers
Variable coefficient DE
There is a y with not a number in front of it.
Separable DE
can be written as
y'=f(y)*g(t)
Integrating Factor
y'+p(t)y=f(t)
Integrating factor: e^(integral p(t) dt)
Only for linear, first order DE!
Superposition for HOMOGENEOUS
If y1 and y2 are solutions then
cy1+ky2 is also a solution
Non-homogeneous principal
y=yh+yp
yh is the general solution to the associated homogeneous DE.
yp is a particular solution to the non-homog. DE.
Solution to
y'=ay+b
ce^(-at)+(b/a)
Exponential growth or decay equation
y'=ky
Solution to y'=ky
y=ce^(kt)
Mixing Problem
x= amount of salt present
(dx)/(dt)=rate in-rate out
rate in=(flow rate in)*(concentration in)
Newton's Law of Cooling
T= temperature of object, M constant temp.
(dT)/(dt)=k(M-T)
Logistic Equation
(dy)/(dt)= (r-ay)y = ry-ay^2
Limit L=r/a
(dy)/(dt)=r(1-y/L)y