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M408D Test 3 (9.1-9.5, 14.1-14.5)
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how do you find new y value using euler's method
y(new)= Y(old)+hy'
what are different about initial value problems
just solve like normal and then use initial values to find the constant
how to find orthogonal trajectories
-solve for k
-differentiate y implicitly in in family one and sub in k value
-solve for y'
-the new dy/dx you are looking for is -1/(y' value just found)
-use separable diff eq techniques to solve and find equation
first order linear diff eq
form: dy/dx + p(x)y = q(x)
y' + (1/x)y = 2
-multiply both sides by I(x) where
i(x) = e^ (int(p(x)dx)
- integrate both sides
-solve for y
function on the left that is prime should be (I(x)y)'
-dont forget + C
how to write domain
D = {(x,y) | x>2, x+y = 1}
three variable:
D= {(x,y,z) (e fork symbol) R^3 | z>y}
little squeeze thrm
make both sides of f(x,y) abs value
separate out part of numerator
set less than or equal to separated out thing
show that the other stuff is less than one
show lim as separated out variable appr 0
when proving a limit dne
try
y=0
x=0
y=x
x=y
y=-x
etc
story problems (mixing)
dy/dx = rate in - rate out
in kg/min (or other v/t)
equation of a plane
linear: ax+by+cz=d
known-point:z-zo=A(x-xo) + B(y-yo)
total differential dz
dz= partialz/partialx(dx) + Partialz/partialy(dy)
dz approximates delta z
use delta x and y for dx and dy
chain rule
dz/dt= parz/parx(dx/dt) + parz/pary(dy/dt)
think tree diagram
implicit diff
dy/dx = -Fx/Fy
logistic model terms
p-population
m=carrying cap
k-growth rate
A-random variable you find
dp/dt= kp(1-p/m)
p(t) = m/(1+Ae^-kt)
a=m-po/po
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