Solutions converge to it
Solutions diverge from it
Solutions converge to it on one side, diverge from it on the other side.
Unstable equilibrium on phase line
Stable equilibrium on phase line
Semistable equilibrium on phase line
Needs Euler method
n, h, starting point, y'
Times in Euler's method
t0= starting t
y in Euler's method
y0= strting y
y1=y0+h*y'(at starting point)
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)
Order of DE
y,y',y'', etc all on first power. (t can be not linear)
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.
can be written as
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
yh is the general solution to the associated homogeneous DE.
yp is a particular solution to the non-homog. DE.
Exponential growth or decay equation
Solution to y'=ky
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.
(dy)/(dt)= (r-ay)y = ry-ay^2