Mr. Integral

THE BIG IDEA

First-Order Methods

Which structure makes a first-order equation solvable?

Classify the equation before manipulating it. Separation, integrating factors, exactness and substitutions each exploit a different structure; phase lines and physical balances keep the algebra connected to behavior.

Begin the first lesson →See the lesson route ↓

6 lessons · No deadlines or locked lessons

SEE THE CONNECTIONS

How the ideas fit together.

A map of the main ideas. Follow a node to its lesson.

  1. Start with01Separation
  2. linear equations instead use02Integrating factors
  3. signs of the rate predict03Equilibrium behavior
  4. balance laws put these into04Physical models

YOUR LEARNING ROUTE

One idea at a time.

Read the explanation, use the visualization, then try the check before opening its reasoning.

Keep the unit cheat sheet handy →
Lessons in this unit6 lessons
  1. 01Separable Equations and Lost EquilibriaSeparate variables while retaining constant solutions and valid intervals.
  2. 02First-Order Linear EquationsBuild an integrating factor and solve a linear initial-value problem.
  3. 03Exact Equations and Potential CurvesRecognize an exact differential and recover the conserved implicit relation.
  4. 04Bernoulli Equations and SubstitutionsConvert a nonlinear power equation to a linear equation without losing valid solutions.
  5. 05Equilibria, Stability and Logistic GrowthUse a phase line to predict long-term behavior and solve a saturation model.
  6. 06Cooling, Mixing and RC CircuitsBuild first-order models from a balance law and interpret their time constant.