Mr. Integral

THE BIG IDEA

Second-Order Linear Equations

How do free motion and forcing combine?

Independent homogeneous solutions describe what a system does on its own. A particular response adds external forcing. Roots, damping and resonance explain the behavior behind the resulting formulas.

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 with01Superposition
  2. characteristic roots construct02Free modes
  3. adding forcing requires03Driven response
  4. sinusoidal forcing reveals04Frequency response

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. 01Superposition and Fundamental SolutionsExplain why two independent solutions determine a second-order homogeneous family.
  2. 02Characteristic Roots and Free ResponseConstruct real solution bases for distinct, repeated and complex roots.
  3. 03Undetermined Coefficients and ResonanceChoose a particular-solution trial and correct it when forcing overlaps a free mode.
  4. 04Damping and Mechanical TransientsConnect characteristic roots to underdamped, critical and overdamped motion.
  5. 05Variation of ParametersConstruct a particular solution for forcing outside the usual trial families.
  6. 06Frequency Response and RLC ModelsCalculate sinusoidal steady-state amplitude and phase, including their assumptions.