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MMLU-Pro / 7879 / A certain spring-mass system satisfies the initial value problem

Problem

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question

A certain spring-mass system satisfies the initial value problem $$ u^{\prime \prime}+\frac{1}{4} u^{\prime}+u=k g(t), \quad u(0)=0, \quad u^{\prime}(0)=0 $$ where g(t)=u3/2(t)−u5/2(t)g(t)=u_{3 / 2}(t)-u_{5 / 2}(t) and k>0k>0 is a parameter. Suppose k=2k=2. Find the time τ\tau after which |u(t)|<0.1|u(t)|<0.1 for all t>τt>\tau.
Plain-text mathematical notation (without MathML)
A certain spring-mass system satisfies the initial value problem
$$
u^{\prime \prime}+\frac{1}{4} u^{\prime}+u=k g(t), \quad u(0)=0, \quad u^{\prime}(0)=0
$$
where g(t)=u_(3/2)(t)−u_(5/2)(t) and k>0 is a parameter.
Suppose k=2. Find the time τ after which |u(t)|<0.1 for all t>τ.
Original LaTeX notation
A certain spring-mass system satisfies the initial value problem
$$
u^{\prime \prime}+\frac{1}{4} u^{\prime}+u=k g(t), \quad u(0)=0, \quad u^{\prime}(0)=0
$$
where $g(t)=u_{3 / 2}(t)-u_{5 / 2}(t)$ and $k>0$ is a parameter.
Suppose $k=2$. Find the time $\tau$ after which $|u(t)|<0.1$ for all $t>\tau$.

category

math

options

  1. 27.8924
  2. 25.6773
  3. 30.7654
  4. 24.5567
  5. 19.4689
  6. 32.8495
  7. 20.5426
  8. 28.6891
  9. 34.1256
  10. 22.3147

src

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Source and history

upstream_id / Original problem ID
7879
task_id
5c693ece-22c5-5552-9311-f1ede613a424
task_key
default--test--7879
task_revision_id
1
config / split
default / test

Official source

initial import