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Form differential models, verify solutions and read direction fields

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Find equilibria before discussing long-time behaviour

For an autonomous equation, constant solutions occur at roots of its rate function. When the rate is the product of y minus 1 and y minus 3, signs are positive below 1, negative between 1 and 3, and positive above 3. Thus 1 attracts nearby solutions and 3 repels them. A direction field gives qualitative local behaviour; existence for arbitrarily large x still needs checking

Year 12, VCE Units 3 & 4, Chapter 6, 1 difficulty level, No calculatorBuild a worksheet

Autonomous equilibria

Constant solutions are roots of the original rate function

  1. Roots Set the rate function to zero and solve for the dependent variable
  2. Signs Inspect g on each interval between roots
  3. Uniqueness State suitable local conditions before ruling out crossing

Worked example

Find and classify the equilibria

\[y'=-(y-1)(y-3)\]
  1. \(\displaystyle y=1,\quad y=3\)These are constant solutions
  2. \(\displaystyle y'<0\ (y<1),\quad y'>0\ (1<y<3)\)The lower root repels nearby solutions
  3. \(\displaystyle y'<0\ (y>3)\)The upper root attracts from both sides

Common mistakes

One point only
✕One zero residual proves a solution✓The residual must vanish throughout an interval
A pointwise coincidence does not establish a differential identity
Tank outflow
✕Use inlet concentration for mass leaving✓Use the current mass divided by the tank volume
The outgoing fluid is assumed perfectly mixed
Zero slope line
✕Every isocline is a solution✓Compare its own derivative with the field
A zero-slope locus can itself have nonzero slope

Difficulty levels

examOnly the original equilibria-signs family

Questions about this topic

Can a double root be treated as two attracting equilibria?+

No. Check signs on both sides and distinguish one-sided attraction

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Specialist Mathematics, chapter 6: Differential equations and rectilinear motion

In this chapter

6A Models, verification and direction fields: Form differential models, verify solutions and read direction fieldsExponential growth and decay rate modelsSeparate variables, impose initial values and retain solution domainsAnti-differentiate a power of a linear functionDifferentiate parametric curves and integrate vector motionSolve and interpret continuous logistic models6C Logistic growth and applications: Form differential models, verify solutions and read direction fieldsApproximate solutions with explicit Euler stepsBinary place value and algorithm tracesUse acceleration forms and differential equations in rectilinear motionAnalyse straight-line motion through displacement, velocity and acceleration