Math Insight

Pure-time differential equation problems

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

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

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

  4. 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 $t$.

      $\displaystyle f{\left (t \right )} =$
    3. Is the population of fish increasing or decreasing at $t=0$?

      Is the rate of change of the population increasing or decreasing at $t=0$?

  5. 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.

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

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

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Math 201, Spring 2015