Math Insight

Pure-time differential equation problems

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  1. Sketch the solution of the pure-time differential equation \begin{align*} \frac{d s}{d t} &= f{\left (t \right )}\\ s(0) &= -1, \end{align*} where $f$ is graphed below.

  2. 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.

  3. 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.
      Need help?
    2. If the population at time $t=0$ is $50$ hundred fish, find an expression for the population at any time $t$.

      $\displaystyle f{\left (t \right )} =$
    3. Determine when the population of fish is increasing and when it is decreasing. Include only times $t \ge 0$.

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

  5. 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.

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

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

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

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

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Math 201, Spring 2016