6 terms

mechanic21PLUS

Radioactive decay series equations

decay rate of nuclear species in this format X->Y->Z is described by:

dx/dt = -λ1*x

dy/dt = λ1*x-λ2*y

dz/dt = λ2*y

dx/dt = -λ1*x

dy/dt = λ1*x-λ2*y

dz/dt = λ2*y

logistic model

x'=αx(K-x), where K is the carrying capacity, and α is a constant

predator-prey model

(by convention, I'm not using c and e for constants, because they don't distinguish integration constants etc for solving DEs and e is not the right constant to use, it's just any old constant, it can be *anything*)

x' = ax-bxy

y' = dxy-fy

or...

x' = x(a-by)

y' = y(dx-f)

Example of one equation:

x'= x(1-x-y)

y'=y((3/4) - x - (1/2)*y)

one outgrows another at a specific point (value)

More information on the constants used in eqn etc (specific constants given on exam):

https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations

x' = ax-bxy

y' = dxy-fy

or...

x' = x(a-by)

y' = y(dx-f)

Example of one equation:

x'= x(1-x-y)

y'=y((3/4) - x - (1/2)*y)

one outgrows another at a specific point (value)

More information on the constants used in eqn etc (specific constants given on exam):

https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations

carrying capacity

In the Logistic Growth Model, what does K stand for?

spring-mass-damper model with no gravity or other forces (note to self-to this def: revision might be needed)

my''+cy'+ky=0, assuming gravity is negligible.

You can solve this as a linear homogeneous DE by plugging in y'=w (1st equation) and solving for the jacobian matrix, etc.

You can solve this as a linear homogeneous DE by plugging in y'=w (1st equation) and solving for the jacobian matrix, etc.

spring-mass-damper model with gravity or other forces (note to self-to this def: revision might be needed)

my''+cy'+ky+f(t)=0, assuming f(t) describes the other forces like gravity or a magnetic force described by this function f(t).

You can solve this as a linear NONhomogeneous DE by plugging in y'=w (1st equation) and solving for the jacobian matrix, etc.

You can solve this as a linear NONhomogeneous DE by plugging in y'=w (1st equation) and solving for the jacobian matrix, etc.