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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.

t1=t0+h etc.

y in Euler's method

y0= strting y

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

etc.

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

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)

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!

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

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.

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

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)

(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)

(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

Limit L=r/a

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