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4PM1/02 Jun 2025, Question 9: The Product and Quotient Rules

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 8 marks

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The question, in short

Given y = e^(−4t) cos 2t, show a given relation involving dy/dt and y, then find integers M and N in a second-order differential equation.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 3 marks

Show that 2e^(−4t) sin 2t equals −dy/dt − 4y.

  1. 1dy/dt = −4e^(−4t) cos 2t − 2e^(−4t) sin 2t
  2. 2−4e^(−4t) cos 2t = −4y
  3. 3dy/dt = −4y − 2e^(−4t) sin 2t, so 2e^(−4t) sin 2t = −dy/dt − 4y

Answer2e^(−4t) sin 2t = −dy/dt − 4y shown

Part (b) · 5 marks

Given d²y/dt² + M dy/dt + Ny = 0 with integers M and N, find M and N.

  1. 1d²y/dt² = −4 dy/dt + 8e^(−4t) sin 2t − 4y
  2. 2From (a): 8e^(−4t) sin 2t = −4 dy/dt − 16y
  3. 3d²y/dt² = −4 dy/dt − 4 dy/dt − 16y − 4y = −8 dy/dt − 20y
  4. 4d²y/dt² + 8 dy/dt + 20y = 0
  5. 5M = 8, N = 20

AnswerM = 8, N = 20

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