Differential Equations

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stable

solutions near it tend towards it t->∞

integrating factor method

µ=>p(t)

Picard's Theorem

1)function is continuous on the region 2)its derivative is continuous on the previous region *it has at least one solution, existance* 3)continuous at its initial value *unique*

Euler's Method Formula

First order

isocline

a curve in the ty plane along which the slope is constant *set of all points where the slope is value is constant

solution

a function that must satisfy the DE for all values of t

direction field

also known as slope field it is the most basic and useful tool of qualitative analysis for first order differential equations *approximate direction of slope at t*

differential equation

an equation that contains the derivatives of one or more dependent variables with respect to tone or more independent variables

ordinary differential equation

contains only ordinary derivatives

partial differential equation

contains partial derivatives

continuous compounding of interest

dA/dt=rA A(t)=A0e^(rt)

newton's law of cooling

dT/dt=k(M-T) M temperature of surrounding T temperature of object

mixing model

dx/dt=rate in- rateout rate=concentration*flowrate

threshold equation

dy/dt=-r(1-y/T)y L is carrying capacity r is initial growth

autonomous

dy/dt=f(y) the independent variable t does not explicitly appear on the right hand side of the equation *no t*

growth and decay

dy/dt=ky y(0)=y0 y(t)=y0e^(kt) k>0 growth k<0 decay

logistic equation

dy/dt=r(1-y/L)*y L is carrying capacity r is initial growth

implicit

equation relating y and t

Power Series Representation

equation y(t)

discretization error

error that results from the process itself (higher h, smaller error)

homogeneous

f(t)=0 equation must end with "equals zero"

existence

model has at least one solution

uniqueness

model has at most one solution

linear

no square roots no squared no sin/cos *can't be inside of something*

Runge-Kutta method

second and fourth order method

equilibrium

set y'=0 when a solution does not change over time it should be a horizontal line

unstable

solutions near it tend away from it t->∞

integrating factor method

solve for µ multiply µ on both sides separate y and t's integrate y=....

initial value problem

the combination of a first order differential equation and an initial condition

round off error

the discrepancy resulting from rounding or chopping numbers at each stage in the computation accuracy *exact representation of fractions* (lower h', lower error)

order of differential equations

the highest-order derivative that appears in the equation

concavity

y''=0 n/a could be inflection point y''<0 concave down y''>0 concave up

Euler-Lagrange Two Stage Method

y'+p(t)y=f(t) yh=ce^-µ yp=v(t)e^-µ yh+yp=y(t)

separable differential equations

y'=f(t)g(y) dy/dt=f(t)g(y) dy/g(y)=f(t)dt *integrals→solve for y→find 'c' with initial values*


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