First order
Of the form dy/dx = f(x)g(y)
What is a separable differential equation?
Form of ODE used for phase plots, and how to find equilibria.
What is autonomous; or dy/dx = f(y). Let dy/dx = 0.
What do equilibrium solutions represent in population problems? And what is always an equilibrium?
The numbers at which a population will stabilise at. x=0 is always stable solution.
Describe the general method to find a solution to an inhomogeneous second order ODE.
1. Set ODE = 0 and trial y=elt to find homogeneous solution 2. a) if homogeneous solution does not match RHS, trial form of RHS and substitute b) if homogeneous solution matches RHS trial form of RHS multiplied by independent variable and substitute. 3) Solve for the constant coefficients.
state the equation of motion for a mass on a spring, and label the coefficients.
mx'' + cx' + kx = f(x)
mass: mass; c: damping coefficient; k: spring constant; f(x): external force
The form of
x dy/dx = x2cosh(5x) + y
What is linear?
For y' = y(2−y), the stability of equilibria 0, 2.
What is unstable, stable?
what is a general form of the ODE for a mixing problem?
dx/dt = (rate in x concentration in) - (rate out x concentration out), where concentration = x/v(t).
y′′ + by′ + 4y = 0 where b ∈ R. For what values of b does the characteristic equation have:
(a) two real distinct solutions; (b) one real repeated solution; (c) two complex conjugate solutions?
(a) |b| > 4 (b) b = ±4 (c) |b| < 4
How can you tell if a spring system is a) underdamped, b) resonant?
a) The characteristic equation gives complex conjugate roots λ=α±iβλ=α±iβ, with α<0 and β≠0, so the solution oscillates with decreasing amplitude.
b) The forcing term oscillates at the same frequency as the system’s natural frequency ω0 (from λ=±iω0 in the undamped characteristic equation), causing a particular solution with a t-factor and growing amplitude.
The solution to
dy/dx = ex-2y
What is y = 0.5log(2ex + C)
construct an ODE dx/dt = f(x), such that if x(0) > 3, the solutions move towards 3, if x(0) < 3 the solution move away from 3.
x(t) = -(x - 3)2
Initially there are x0 bacteria. Assume that the rate of growth of the bacteria is proportional to the population x at time t. After 1 hour there are 1.5x0 bacteria. Find x(t).
x(t) = x0 (1.5)t
find general solution of y′′ + 2y′ + y = 3x2 + 2
y(x) = (A + Bx)e-x + 3x2 − 12x + 20
A 10 kilogram mass is suspended from the end of a vertical spring with spring constant 180 N/m. At time t = 0, the mass is started in motion from the equilibrium position with an initial velocity of 1 m/s in the upward direction. At the same time, a constant downward force of 360 Newtons is applied to the system. Assume that air resistance acts with damping constant β = 60 N/(m/s). Let x be the displacement in metres of the mass below the equilibrium position at time t seconds. Find the position of the mass at any time t ≥ 0 and identify the transient and steady-state parts of your solution.
x'' + 6 x' + 18x = 36 ->
x(t) = −2e−3t cos(3t) − 7/3 e−3t sin(3t) + 2
first two terms transient, last term steady state
The solution to
xy′ + y = esinx cos x, y(pi) = 1
what is y = (esinx + π − 1) / x
dy/dt = y(2 − y)(y − 1); for what values of y(0) will lim(t→∞) y(t) = 2? Draw phase plot and family of solutions.
y(0) > 1
A river flows into a lake initially containing 100 (km)3 of pure water at a rate of 48 (km)3/year. Pollutant from a factory is released into the lake at the rate of 2 (km)3/year. Assume that the pollutant mixes immediately and uniformly with the rest of the water in the lake. The lake also flows into another river at the rate of 40 (km)3/year. Find dx/dt.
dx/dt + 4x/(10 + t) = 2
find the general solution of y′′ − 7y′ + 10y = e2x where y(0) = 4 and y′(0) = 50/3
y(x) = (1 − 1/3 x)e2x + 3e5x
A mass of 0.5 kg is attached to a spring suspended from the ceiling of a room. The spring constant is 5 N/m. Assume the damping force acts with damping constant β ≥ 0. Let y metres be the displacement of the mass below the equilibrium position at time t seconds. An external force of sin(ωt) N is applied to the system, where ω ≥ 0. The system is set into motion by pulling the mass down 0.3 m below its equilibrium position and releasing it from rest.
for which values of β and ω will the displacement of the mass be unbounded (resonance)? extension: find x(t) in that case.
to have resonance, we need β = 0 and ω = √10.
y(t) = 3/10 cos√10 t? + 1/10 sin√10 t? − 1/√10 t cos√10 t?.