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Equilibrium Solution

y'=0

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Stable Equilibrium

Solutions converge to it

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Unstable Equilibrium

Solutions diverge from it

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Semistable Equilibrium

Solutions converge to it on one side, diverge from it on the other side.

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Source (repeller)

Unstable equilibrium on phase line

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Sink (attractor)

Stable equilibrium on phase line

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Node

Semistable equilibrium on phase line

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Needs Euler method

n, h, starting point, y'

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Times in Euler's method

t0= starting t

t1=t0+h etc.

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y in Euler's method

y0= strting y

y1=y0+h*y'(at starting point)

etc.

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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

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Order of DE

Highest derivative

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Linear DE

y,y',y'', etc all on first power. (t can be not linear)

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Homogeneous DE

The term without any y is 0.

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Constant coefficient DE

In front of the ys we have numbers

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Variable coefficient DE

There is a y with not a number in front of it.

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Separable DE

can be written as

y'=f(y)*g(t)

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Integrating Factor

y'+p(t)y=f(t)

Integrating factor: e^(integral p(t) dt)

Only for linear, first order DE!

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Superposition for HOMOGENEOUS

If y1 and y2 are solutions then

cy1+ky2 is also a solution

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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.

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Solution to

y'=ay+b

ce^(-at)+(b/a)

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Exponential growth or decay equation

y'=ky

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Solution to y'=ky

y=ce^(kt)

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Mixing Problem

x= amount of salt present

(dx)/(dt)=rate in-rate out

rate in=(flow rate in)*(concentration in)

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Newton's Law of Cooling

T= temperature of object, M constant temp.

(dT)/(dt)=k(M-T)

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Logistic Equation

(dy)/(dt)= (r-ay)y = ry-ay^2

Limit L=r/a

(dy)/(dt)=r(1-y/L)y

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