Math Insight

Pure-time differential equation problems

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  1. Find all solutions to the pure-time differential equation $\frac{d p}{d t} = 4 e^{- 4 t}$.
    $p{\left (t \right )}=$

    For full credit, do not round your answer to a decimal. (Maximum of 80% credit will be awarded for rounded answers that include at least 5 correct significant digits.)

  2. Solve the pure-time differential equation $\frac{d h}{d t} = 7 e^{4 t}$ with initial condition $h{\left (3 \right )} = -1$.
    $h{\left (t \right )}=$

    For full credit, do not round your answer to a decimal. (Maximum of 80% credit will be awarded for rounded answers that include at least 5 correct significant digits.)

  3. Solve the pure-time differential equation $\frac{d y}{d t} = - 5 t^{7} + 10 t^{4} - 5 t^{2}$ with initial condition $y{\left (0 \right )} = 0$.
    $y{\left (t \right )}=$

    For full credit, do not round your answer to a decimal. (Maximum of 80% credit will be awarded for rounded answers that include at least 5 correct significant digits.)

  4. Solve the pure-time differential equation $\frac{d y}{d t} = 10 e^{4 t}$ with initial condition $y{\left (0 \right )} = -8$.
    $y{\left (t \right )}=$

    For full credit, do not round your answer to a decimal. (Maximum of 80% credit will be awarded for rounded answers that include at least 5 correct significant digits.)

  5. Find all solutions to the pure-time differential equation $\frac{d g}{d t} = 2 t^{5} + 8 t^{2} + 2$.
    $g{\left (t \right )}=$

    For full credit, do not round your answer to a decimal. (Maximum of 80% credit will be awarded for rounded answers that include at least 5 correct significant digits.)

  6. The population $f{\left (t \right )}$ of fish, measured in hundreds, is growing at a rate of $e^{- 3 t}$ hundred/month.
    1. Write a differential equation modeling this scenario.


    2. If the population at time $t=0$ is $50$ hundred fish, find an expression for the population at any time $t$.

      $\displaystyle f{\left (t \right )} =$

      For full credit, do not round your answer to a decimal. (Maximum of 80% credit will be awarded for rounded answers that include at least 5 correct significant digits.)
    3. Determine when the population of fish is increasing and when it is decreasing. Include only times $t \gt 0$.
      The population is increasing for
      .
      The population is decreasing for
      .
      In each blank enter an inequality in terms of $t$, such as t > 5 or 3 < t < 5. If the answer is all time, enter t > 0. If the answer is never, enter never in the answer blank (even though it doesn't make grammatical sense for the sentence).

  7. Use Forward Euler with time steps of size $\Delta t = 0.5$ to approximate the solution of the differential equation $\frac{d w}{d t} = 6 \ln{\left (3 t^{4} + 3 \right )}$ at time $t=2$, given that $w{\left (0 \right )}=5$. If rounding, be sure to include at least $5$ significant digits. (To be safe, include even more digits.)


  8. Sketch the solution of the pure-time differential equation \begin{align*} \frac{d x}{d t} &= h{\left (t \right )}\\ x(0) &= 5, \end{align*} where $h$ is graphed below. (All values of $t$ where $h(t)=0$ are integers.)

    Use the below applet to graph the $x(t)$. Use the slider $n_c$ to specify the number of critical points of $x(t)$. Move the right two points to specify the shape of the graph of $x(t)$. Move the left point up and down to shift the graph vertically. (If the left point is not visible, you can move the right two points until left part of the graph, and hence the left point, becomes visible.)

    Feedback from applet
    Critical point locations:
    Increasing/decreasing over correct intervals:
    Initial condition:
    Number of critical points:

    The applet will grade your graph based on the initial condition and intervals where $x(t)$ is increasing and decreasing. Getting the precise shape of the graph of $x(t)$ is not important (and may be impossible to achieve with the applet).

  9. Use Forward Euler with time steps of size $\frac{1}{2}$ to approximate the solution of the differential equation $\frac{d z}{d t} = - 8 t^{2} - 6 t$ at time $t=2$, given that $z{\left (0 \right )}=-6$. If rounding, be sure to include at least $5$ significant figures.