臺灣大學數學系
八十七學年度碩士班甄試入學考試試題 微分方程
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Choose 2 Problem from below. 15 points each.
Find the general solutions of the following systems 1.
2.
1.
Assume satisfies
Show that and .
1.
Show that if satisfies
then . 2.
2.
Consider a particle sliding on a frictionless curve given by under the influence of gravity, where is the arc length of the curve. Let be the time.
Show that the motion of the particle satisfies the equation
where is the gravitational acceleration.
1.
3.
Assume when . Show that and
for some constant . 2.
Let be a cycloid
with parameter . Show that a particle oscillate on has the period of oscillation .
3.
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