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First-Order Linear and Logistic Equations worksheets

Use integrating factors for linear IVPs and solve or interpret bounded logistic models.

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Use integrating factors for linear IVPs and solve or interpret bounded logistic models.

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Solutions include their mathematical boundaries

Worked keys preserve equilibrium branches, maximal intervals, recurrence rounding, integrating factors, and carrying-capacity interpretations wherever they apply.

Learning objective

What this activity builds

Derive linear and logistic formulas while identifying equilibria, units, and long-term behavior.

Worked example

What belongs in a complete differential-equation solution?

The governing equation, valid solution steps, initial-condition substitution, and any interval, units, equilibrium, or approximation label.

The worked key keeps the method and interpretation visible instead of presenting only a final expression.

Teaching notes

Make the practice count

  • Check a symbolic solution by substitution into the differential equation.
  • Keep exact formulas separate from Euler approximations.
  • Ask what the result means on the original interval and in the stated units.

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Math SheetNameScoreTeacherDateFirst-Order Linear and Logistic EquationsFor linear equations, show the integrating factor. For logistic equations, give the solution formula, equilibria, units, and long-term behavior.1.Let P(t) be a population in organisms, with t in years. Solve P' = 1/2P(1-P/180), P(0) = 30; state both equilibria and long-term behavior.2.A population P(t), measured in organisms with t in years, follows P' = 1/4P(1-P/180) and P(0) = 90. Find P(t), both equilibria, and the long-termpopulation.Show recurrence rows, integrating factors, equilibria, intervals, and units when applicable.math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateFirst-Order Linear and Logistic EquationsFor linear equations, show the integrating factor. For logistic equations, give the solution formula, equilibria, units, and long-term behavior.3.A concentration C(t), in mg/L, satisfies C' + 3C = 12, C(0) = 7. Find the concentration and its long-term value.4.Solve y' + 3y = 9, y(1) = 5. Use an integrating factor and state the maximal interval.Show recurrence rows, integrating factors, equilibria, intervals, and units when applicable.math-sheet.comPage 2 of 2

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