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Unit 3: Existence, Uniqueness, and Qualitative Behavior

When a solution is guaranteed to exist and be the only one, and how a phase line predicts a solution's fate without ever solving the equation.

8 topics · 29 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

2.8 Existence and Uniqueness for First-Order Equations

  1. How it worksHow do continuity conditions on f and its partial derivative guarantee one solution through a starting point?
  2. Common mistakeWhy doesn't the existence and uniqueness theorem guarantee a solution exists for all future time?
  3. Going furtherWhat happens to a solution's guaranteed interval of existence as the starting point moves closer to where f blows up?
  4. Worked exampleShow that the equation y' = y2, y(0) = 1 satisfies the existence and uniqueness theorem's hypotheses near t = 0.
Your turn1 of 2
For y' = y2, y(0) = 2, the exact solution is y = y0/(1 - y0 t). Find y(t) at t = 0.23.

2.8 Why Solution Curves of the Same Equation Never Cross

  1. How it worksHow does uniqueness at every point force two nearby solution curves to stay separated forever?
  2. Worked exampleShow that y = ex and y = ex + 1 never cross by comparing them as solutions of y' = y.
  3. Going furtherWhat would it mean physically if two different cooling curves for the same object actually did cross?
Your turn1 of 2
For y' = y, compare the two solutions y1 = 3ex and y2 = 8ex. Find the value of y2(x) - y1(x) at x = 1.61.

2.8 When the Uniqueness Theorem's Conditions Fail

  1. How it worksHow does a vertical tangent in y1/3 at zero break the smoothness the uniqueness theorem needs?
  2. Worked exampleShow that both y = 0 and y = (2x/3)3/2 solve y' = y1/3 with y(0) = 0.
  3. Common mistakeWhy doesn't finding one valid solution to an initial value problem guarantee it's the only one?
Your turn1 of 2
The nonzero solution branch of y' = 3y1/3, y(0) = 0, is y = ((2k/3)x)1.5. Find y(2.12).

2.5 Autonomous Equations and Equilibrium Solutions

  1. How it worksHow do you find the equilibrium solutions of an autonomous equation just by looking at where the rate is zero?
  2. Worked exampleFind the equilibrium solutions of dy/dt = y(y - 3)(y + 2).
  3. Going furtherWhat happens to a solution of an autonomous equation that starts exactly on one of its equilibrium values?
  4. Common mistakeWhy doesn't an autonomous equation's solution curve look identical no matter which two equilibria it starts between?
Your turn1 of 2
Find the sum of the three equilibrium solutions of dy/dt = y(y - 5)(y + 5).

2.5 The Phase Line and Flow Arrows

  1. How it worksHow do the signs of f(y) between consecutive equilibria turn into arrows pointing up or down the phase line?
  2. How it worksHow does reading the arrows on either side of an equilibrium tell you what nearby solutions do over time?
  3. Going furtherWhat happens on the phase line when two equilibria merge into one as a parameter changes?
Your turn1 of 2
For dy/dt = (y - 2)(y - 8), find the value of dy/dt at y = 5.

2.5 Stable, Unstable, and Semistable Equilibria

  1. How it worksHow do the arrows on both sides of an equilibrium determine whether it's stable, unstable, or semistable?
  2. Worked exampleFind the stability of each equilibrium of dy/dt = y(1 - y)(y - 2).
  3. Common mistakeWhy doesn't a stable equilibrium attract every single solution on the entire phase line?
  4. Going furtherWhat happens to a solution that starts just barely on the unstable side of a semistable equilibrium?
Your turn1 of 2
For dy/dt = (2 - y)(y - 8) with a < b, the right-hand side is f(y). Find f'(2), the derivative of f evaluated at the equilibrium y = a. (A positive value means solutions move away — the equilibrium is unstable there.)

2.5 The Logistic Equation's Phase Line

  1. How it worksHow do the two equilibria at 0 and the carrying capacity K show up on the logistic equation's phase line?
  2. Worked exampleFind the equilibria and their stability for dP/dt = 0.5P(1 - P/200).
  3. Going furtherWhat happens to the logistic phase line when the carrying capacity K is cut in half?
  4. Common mistakeWhy doesn't a population starting above the carrying capacity K keep growing past it instead of shrinking back down?
Your turn1 of 2
For the logistic equation dP/dt = 0.687P(1 - P/360), find the value of dP/dt when P = 360/2.

2.5 Bifurcations as a Parameter Changes

  1. How it worksHow does plotting equilibria against a changing parameter build a bifurcation diagram?
  2. Worked exampleFind the bifurcation value of h in dy/dt = y2 - h and describe what happens to the equilibria as h crosses it.
  3. Going furtherWhat happens to the phase line just before and just after a saddle-node bifurcation occurs?
  4. Ties togetherHow does a bifurcation diagram's curve of equilibria connect to the phase lines you'd draw at each fixed parameter value?
Your turn1 of 2
Find the bifurcation value h·of the parameter h in dy/dt = y2 - 5y - h, the value at which the number of equilibria changes.