The deflection of a diving board

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Ordinary differential equations

Mathematical Methods for Physics, Ch Higher-order linear equations, the Wronskian and variation of parameters

Problem

The deflection

y(t)

of a diving board, whose effective stiffness changes as a diver walks along it, obeys

(\tau^2+t^2)\,y'' + 4t\,y' + 2y = 0, \qquad t \ge 0\,\mathrm{s},

where

y

is measured in centimetres,

t

in seconds, and

\tau = 1.0\,\mathrm{s}

sets how quickly the diver's walk changes the stiffness. Two independent solutions

y_1(t)

and

y_2(t)

of this equation have Wronskian

W(y_1,y_2)(0\,\mathrm{s}) = 6.0\,\mathrm{cm^2\,s^{-1}}

. ((a)) Write the equation in standard form and identify

a_1(t)

. ((b)) Use Abel's identity to obtain

W(y_1,y_2)(t)

as a function of

t

, without solving the equation. ((c)) Evaluate

W(y_1,y_2)(3.0\,\mathrm{s})

Related theory

Ordinary differential equations

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