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MathidiaMathematics Academy
Year 2 mathematical synthesis
Lesson 75 of 75
Review

Year 2 review

Objective. Solve unfamiliar second-year core problems.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Year 2 review draws together the course's core strands, optimisation, mechanics, PDEs and modelling, requiring flexible recognition of which technique a problem demands. Reviewing means practising rapid identification: does a question need a Lagrangian, a separation-of-variables PDE solution, or a dimensional argument, before carrying out the correct calculation. Example: a problem describing a constrained oscillating system might need generalised coordinates and Lagrange's equations together, while one about diffusion needs separation of variables and eigenfunctions, so review practice should mix problem types unpredictably. Strong exam performance comes from this recognition skill as much as from computational fluency.

Worked example

As a Year 2 review problem, a particle of mass m moves on a frictionless line with Lagrangian L = ½mẋ² − ½kx². Find the angular frequency ω of oscillation implied by its equation of motion.

2
Illustrated card showing 2
  1. 1Apply Lagrange's equation as in analytical mechanics: d/dt(∂L/∂ẋ) − ∂L/∂x = 0
  2. 2Compute terms: mẍ − (−kx) = 0, giving mẍ + kx = 0
  3. 3Compare to standard form ẍ + ω²x = 0, so ω² = k/m
  4. 4Take square root: ω = √(k/m)

Answer: ω = √(k/m)

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

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Common mistake. Students revise each topic in isolation and then struggle on mixed exam papers; practising unsorted, mixed-topic problem sets is essential to train quick recognition of which method a given question needs.

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