# Test 1

### 26 terms by diffeq2012

#### Study  only

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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'

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.

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

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

Example: