test 2 Flashcards
if you can’t solve a homogenous DE, what should you substitute
For first order homogeneous DEs.
v=y/x. Transforms it into a separable DE.
use implicit on this substitution eqn since youll probably need v’
how do you know if a DE is homogeneous?
if there are the same amount of indep and dep variables in the numerator and the denominator respectively
definition of bernoulli
Standard Form: y’ + P(x)y = Q(x)y^n ,<- usually a weird nonlinear piece.
how do you solve a bernoulli DE?
make the substitution v = y^(1-n) after you get the DE into standard form. You should be able to transform this into a linear DE.
what’s the checklist you should run thru when choosing a solution strat?
- is it first order linear?
- is it separable
- is it homogeneous
- is it bernoulli?
what does an asympotically stable sink or attractor?
on a phase portrait, it’s an equilibrium point where both arrows point toward it
what does an unstable source or repeller look like?
on a phase portrait, it’s an equilibrium point where both arrows point away from it.
what does a semi stable node look like?
on a phase portrait, it’s an equilibrium point where one point points toward and one arrow points away
how do you create a phase portait w/o the derivative test?
find your equilibruim points of a single variable (autonomous) DE, then find the signs of points in b/w those equilibruim points
how do create a phase portrait with the derivative test?
find your equilibruim points then take the derivaitive of the function at those points. if the derivative is pos then it’s unstable, neg then it’s stable, equal to 0 then no conculsion
other names for equilibrium point
critical point, stationary point, fixed point or steady state
how would you turn a phase portrait into a graph?
edge the solution lines to equilibrium points. the equilibrium points would be straight lines b/c the derivative at those points are 0
how do you predict with eulers method by hand?
y_t = y_0 + f(x_0, y_0) + step size (h). Repeat as many times as it asks
