 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-2/13c/b/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Integrating the ordinary differential equation four times gives a quartic polynomial. It is useful to use the linearity of the equation and split the solution into a gravity part and a force part:

y = y_{0} + y_{F},

where

y_{0} (x) = - \frac{ρ g}{24 A} x^{2} (6 L^{2} - 4 Lx + x^{2})

satisfies the clamped and torque-free boundary conditions with

F = 0

, while

y_{F} (x) = \frac{F}{6 A} x^{2} (3 L - x)

satisfies the homogeneous equation and contributes the endpoint force. Direct differentiation verifies

y (0) = y^{'} (0) = 0, y^{''} (L) = 0, - A y^{'''} (L) = F .

Thus

y (x) = - \frac{ρ g}{24 A} x^{2} (6 L^{2} - 4 Lx + x^{2}) + \frac{F}{6 A} x^{2} (3 L - x),

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