A-Level Differential equations: worked solution

Question

A social-media trend's popularity P at time t days (0 <= t <= 100) satisfies d^2P/dt^2 = -0.05(dP/dt)^2. Using v = dP/dt, show v = 20/(t + C) for a real constant C.

Step 1: use the substitution to reduce the order of the equation. Let v = dP/dt. Differentiating once more, d²P/dt² = dv/dt. The given equation d²P/dt² = -0.05(dP/dt)² therefore becomes a first-order equation in v: dv/dt = -0.05v² Step 2: separate the variables. Assuming v ≠ 0, divide both sides by v² and multiply by dt: (1/v²) dv = -0.05 dt Integrate both sides: ∫ v⁻² dv = ∫ -0.05 dt Step 3: do the two integrals. On the left, ∫ v⁻² dv = v⁻¹/(-1) = -1/v. On the right, ∫ -0.05 dt = -0.05t + c', where c' is the constant of integration. So -1/v = -0.05t + c' Step 4: make v the subject. Multiply both sides by -1: 1/v = 0.05t - c' Take reciprocals: v = 1/(0.05t - c') Multiply the numerator and denominator by 20 to clear the decimal (0.05 × 20 = 1): v = 20/(20 × 0.05t - 20c') = 20/(t - 20c') Step 5: rename the constant. -20c' is just some real number; call it C. Then v = 20/(t + C) (as required)