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MMLU-Pro / 8638 / Consider the initial value problem

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

Consider the initial value problem $$ y^{\prime \prime}+\gamma y^{\prime}+y=k \delta(t-1), \quad y(0)=0, \quad y^{\prime}(0)=0 $$ where kk is the magnitude of an impulse at t=1t=1 and γ\gamma is the damping coefficient (or resistance). Let γ=12\gamma=\frac{1}{2}. Find the value of kk for which the response has a peak value of 2 ; call this value k1k_1.
Plain-text mathematical notation (without MathML)
Consider the initial value problem
$$
y^{\prime \prime}+\gamma y^{\prime}+y=k \delta(t-1), \quad y(0)=0, \quad y^{\prime}(0)=0
$$
where k is the magnitude of an impulse at t=1 and γ is the damping coefficient (or resistance).
Let γ=(1)/(2). Find the value of k for which the response has a peak value of 2 ; call this value k₁.
Original LaTeX notation
Consider the initial value problem
$$
y^{\prime \prime}+\gamma y^{\prime}+y=k \delta(t-1), \quad y(0)=0, \quad y^{\prime}(0)=0
$$
where $k$ is the magnitude of an impulse at $t=1$ and $\gamma$ is the damping coefficient (or resistance).
Let $\gamma=\frac{1}{2}$. Find the value of $k$ for which the response has a peak value of 2 ; call this value $k_1$.

category

math

options

  1. 3.1415
  2. 3.9022
  3. 2.8108
  4. 2.0000
  5. 3.5672
  6. 2.3456
  7. 4.0000
  8. 2.7182
  9. 1.7890
  10. 1.6180

src

scibench-diff

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Source and history

upstream_id / Original problem ID
8638
task_id
1380ad1b-9c8f-5a71-9bc4-94b1eea38912
task_key
default--test--8638
task_revision_id
1
config / split
default / test

Official source

initial import