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Order of a differential equation
Give order for these equations: [image] |
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| Highest derivative it has (ex y'' = second order) |
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A linear equation doesn't mix and match y and its derivatives (ex first derivative only contains y' term)
A non linear doesnt (ex - the term y * y'') |
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| Find where a solution is continuous |
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Definition
| Find where variable (example - t) cannot equal 0. Then plug in initial condition to see which interval is valid. |
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| What is an autonomous equation? |
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| d/dx(1/tan(x)) = (both in terms of cxx and answer) |
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| d/dx cot(x) = -csc^2(x) = -1/sin^2(x) |
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| How to reduce second order to first order? |
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| Model free-fall of mass m and drag coefficient b |
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Definition
[image]
k -> constant
u -> temperature of object
T -> ambient (room) temprature
t -> independent time variable |
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Chemical Concentration Problem
A pond initially contains 1,000,000 gal of water and an unknown amount of an undesirable chemical. Water containing 0.04 g of this chemical per gallon flows into the pond at a rate of 600 gal/hr. The mixture flows out at the same rate, so the amount of water in the pond remains constant. Assume that the chemical is uniformly distributed throughout the pond. |
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Definition
dQ/dt = rate in - rate out
rate in = 0.04 grams / gallon * 600 gallon/hour
rate out = (unknown)/1,000,000 gallons * 600 gal/hr
General Steps ->
Find concentration on grams/hr - and convert that. |
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| Stable, Semistable, Unstable |
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Definition
Stable V^
Semistable ^^ OR VV
Unstable V^ |
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Definition
| dv/dt = dv/dx*dx/dt = dv/dx * v |
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| Say the Bernoulli term is y^4. Find v' in terms of y' |
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Definition
v = y^(1-n)
v = y^-3
v' = -3y^-4*y'
Solve and substitute |
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Logistic Growth Model
Now consider a species of fish: y(t) = population (in millions), t = time (in years),r = 8 per year is the intrinsic growth rate, K = 4 million is the carrying capacity, and annual catch = 6 million |
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Definition
[image]
Plug it in to see the result. The above can be modified for harvesting as
[image] - harvest
The answer to the fish question is
[image] |
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| Second order - t missing - how to solve |
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Definition
1) get it in the form y''= f(y,y')
2)Replace y'' with [image] on the left side
3) Set v = y'. Replace every y' on the right with v
4) Solve the above for v. v = dy/dt
5) Solve for y again. |
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Definition
sin(2x) = 2sin(x)cos(x)
cos(2x) = cos^2(x) - sin^2(x) |
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Definition
sin(2x) = 1/2(1-cos(2x)
cos(2x) = 1/2(1+cos(2x) |
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