First-Order DEs
Method used to solve the DE: y' = M(x)N(y).
What is separation of variables?
A ___ occurs at the value of a parameter where the number of equilibrium solutions changes.
What is a bifurcation?
dx/dt = f(x); dy/dt = g(y)
What is a decoupled system of DEs?
K in the population model dP/dt = r(1-P/K)P
What is the carrying capacity?
h in Euler's Method formula: Yk+1 = Yk + hf(Yk, tk)
x' + p(t)x = q(t)
What is a first-order linear differential equation?
Solutions to a differential equation y' = f(y) given by f(y) = 0 for all y.
What are equilibrium solutions?
Rewrite x'' + 3x' + 2x = 0 as a system of differential equations.
What is x' = v; v' = -2x - 3v?
Suppose we are trying to model the population of lynx and snowshoe hares. We assume that in the absence of lynx, the snowshoe hare population will reproduce at a rate proportional to their population and are not affected by overcrowding (and also don't have any other predators!) Please write the differential equation for the hare population.
What is dH/dt = aH?
The interval which we are guaranteed to have a unique solution to an IVP on.
What is interval of validity?
Solution to the differential equation: dy/dx = 6y^2x, y(1) = 1/25
What is y(x) = 1/(28 - 3x2)?
An equilibrium solution such that all nearby solutions tend to converge towards it over time.
What is a sink?
x''' + x'' + 2x = 0 is a ___ order DE.
What is third?
Suppose we are trying to model the population of lynx and snowshoe hares. We assume that in the absence of lynx, the snowshoe hare population will reproduce at a rate proportional to their population and are not affected by overcrowding (and also don't have any other predators!) Now, assume that the rate at which hares are consumed by the lynx is proportional to the rate at which hares and lynx interact. Please write the differential equation for the hare population.
What is dH/dt = aH - bHL?
Often in Sage, we use this numerical method to solve our IVPs. This method is more accurate than Euler's Method.
What are Runge-Kutta or RK methods?
The solution to the IVP:
y' - (2/x)y = 2x2, y(-2) = 4.
What is y(x) = 2x3+5x2?
The x-nullcline(s) of the system:
x' = x+y
y' = -2x+y
What is: y = -x?
Please find the solution to the partially couple system:
x' = 2x
y' = x + 3y
What is x = c1e2t; y = -c1e2t + c2e3t?
A 1500 gallon tank initially contains 600 gallons of water with 5 lbs of salt dissolved in it. Water enters the tank at a rate of 9 gal/hr and the water entering the tank has a salt concentration of 15(1+cos(t)) lbs/gal. If a well mixed solution leaves the tank at a rate of 6 gal/hr, how much salt is in the tank at t = 300 hours?
What is Q(300)=279.797 lbs?
Two conditions that need to be met to guarantee a unique solution to an IVP (y' = f(t, y), y(0) = t_0).
What is f(t, y) needs to be continuous at the initial value and partial of f with respect to y needs to be continuous at the initial value.
The value of k in the DE: P' = kP, where P(0) = 1000 and P(1) = 1030.
What is k = 0.0296?
The equilibrium points of the system:
x' = x(1-x-2y)
y' = y(1-y-2x)
What are: (0, 1); (1/3, 1/3); (1,0); (0,0)?
Suppose that we have two tanks (A and B) between which a mixture of brine flows. Tank A contains 300 liters of water in which 100 kilograms of salt has been dissolved and Tank B contains 300 liters of pure water. Fresh water is pumped into Tank A at the rate of 500 liters per hour, and brine is pumped into Tank B from Tank A at the rate of 500 liters per hour. Brine is also drained at a rate 500 liters of brine per hour from Tank B. All brine mixtures are well-stirred. If we let x(t) be the amount of salt in Tank A at time t and y(t) be the amount of salt in Tank B at time t, please set up the IVP for this scenario.
x'(t) = -5/3 x; y'(t) = 5/3 x - 5/3 y; x(0) = 100; y(0) = 0
A population of insects in a region will grow at a rate that is proportional to their current population. In the absence of any outside factors the population will triple in two weeks time. On any given day there is a net migration into the area of 15 insects and 16 are eaten by the local bird population and 7 die of natural causes. There are initially 100 insects in the population. Please set up the IVP for this scenario.
What is P' = (ln3/14)P - 8, P(0) = 100?
Determine the interval of validity for the initial value problem below and give its dependence on the value of y0
y' = y2; y(0) = 2
What is -infinity to 1/2?