Differential Equations
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*
