A quantity , measured in metres, obeys y - 2 t y + 2 t^2 y = g, t t_0, with constant…

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

A quantity

y(t)

, measured in metres, obeys

\ddot{y} - \frac{2}{t}\dot{y} + \frac{2}{t^2}y = g, \qquad t \ge t_0,

with

g = 2.0\ \mathrm{m\,s^{-2}}

constant and

t_0 = 1.0\ \mathrm{s}

. It is at rest at

t_0

y(t_0) = 0

and

\dot{y}(t_0) = 0

. Substitution shows that

y = t

solves the undriven equation; no other solution is given. ((a)) Find

y(t)

for

t \ge t_0

, and evaluate

y(2.0\ \mathrm{s})

to three significant figures. ((b)) Say which tools the answer needed, and why Chapter 8's route (an auxiliary equation for the undriven solutions, then a trial form guessed from the drive) is not available here, even though the drive is a constant.

Related theory

Ordinary differential equations

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