Math Insight

Forward Euler and linear approximations

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  1. For the following differential equation \begin{align*} \diff{ u }{ t } &= - 0.1 u - 9.2\\ u(0) & = -8.3 \end{align*} calculate a Forward Euler approximation to $u(21.0)$ using a time step of $\Delta t = 7.0$. In this problem, we guide your through the steps of the calculation. It may easier to understand the steps by writing the differential equation using an explicit argument of $t$ \begin{align*} \diff{ u }{ t }(t) &= - 0.1 u{\left (t \right )} - 9.2. \end{align*}

    1. The following steps will guide you in calculating the Forward Euler approximation after one step of size $\Delta t = 7.0$, yielding an estimate of $u(7.0)$.

      What is the initial condition? $u(0) = $

      Based on that result and the differential equation, what is the slope of $u(t)$ at time $t=0$?
      $\diff{ u }{t}(0) = - 0.1 u{\left (0 \right )} - 9.2 =$

      Those two numbers are enough information to make a linear approximation at $t=0$.
      $L(t) = u(0) + \diff{ u }{t}(0) (t-0) =$

      We use this linear approximation to approximate the value of $u(t)$ at $t=\Delta t = 7.0$.
      $u(7.0) \approx L(7.0) =$

    2. The following steps will guide you in calculating the Forward Euler approximation after the second step of size $\Delta t = 7.0$, yielding an estimate of $u(14.0)$.

      Given the value of $u(7.0)$ estimated above and the differential equation, what is the slope of $u(t)$ at time $t=\Delta t = 7.0$?
      $\diff{ u }{t}(7.0) = - 0.1 u{\left (7.0 \right )} - 9.2 =$

      Those two numbers are enough information to make a linear approximation at $t=\Delta t = 7.0$.
      $L(t) = u(7.0) + \diff{ u }{t}(7.0) (t-7.0) =$

      We use this linear approximation to approximate the value of $u(t)$ at $t=2\Delta t = 14.0$.
      $u(14.0) \approx L(14.0) =$

    3. The following steps will guide you in calculating the Forward Euler approximation after the third step of size $\Delta t = 7.0$, yielding an estimate of $u(21.0)$.

      Given the value of $u(14.0)$ estimated above and the differential equation, what is the slope of $u(t)$ at time $t=2\Delta t = 14.0$?
      $\diff{ u }{t}(14.0) = - 0.1 u{\left (14.0 \right )} - 9.2 =$

      Those two numbers are enough information to make a linear approximation at $t=2\Delta t = 14.0$.
      $L(t) = u(14.0) + \diff{ u }{t}(14.0) (t-14.0) =$

      We use this linear approximation to approximate the value of $u(t)$ at $t=3\Delta t = 21.0$.
      $u(21.0) \approx L(21.0) =$

  2. For the following logistic differential equation \begin{align*} \diff{ v }{ t } &= 0.3 v \left(- \frac{v}{170} + 1\right)\\ v(0) & = 187 \end{align*} calculate a Forward Euler approximation to $v(9.9)$ using a time step of $\Delta t = 3.3$. In this problem, we guide your through the steps of the calculation. It may easier to understand the steps by writing the differential equation using an explicit argument of $t$ \begin{align*} \diff{ v }{ t }(t) &= 0.3 \left(- \frac{1}{170} v{\left (t \right )} + 1\right) v{\left (t \right )}. \end{align*}

    1. The following steps will guide you in calculating the Forward Euler approximation after one step of size $\Delta t = 3.3$, yielding an estimate of $v(3.3)$.

      What is the initial condition? $v(0) = $

      Based on that result and the differential equation, what is the slope of $v(t)$ at time $t=0$?
      $\diff{ v }{t}(0) = 0.3 \left(- \frac{1}{170} v{\left (0 \right )} + 1\right) v{\left (0 \right )} =$

      Those two numbers are enough information to make a linear approximation at $t=0$.
      $L(t) = v(0) + \diff{ v }{t}(0) (t-0) =$

      We use this linear approximation to approximate the value of $v(t)$ at $t=\Delta t = 3.3$.
      $v(3.3) \approx L(3.3) =$

    2. The following steps will guide you in calculating the Forward Euler approximation after the second step of size $\Delta t = 3.3$, yielding an estimate of $v(6.6)$.

      Given the value of $v(3.3)$ estimated above and the differential equation, what is the slope of $v(t)$ at time $t=\Delta t = 3.3$?
      $\diff{ v }{t}(3.3) = 0.3 \left(- \frac{1}{170} v{\left (3.3 \right )} + 1\right) v{\left (3.3 \right )} =$

      Those two numbers are enough information to make a linear approximation at $t=\Delta t = 3.3$.
      $L(t) = v(3.3) + \diff{ v }{t}(3.3) (t-3.3) =$

      We use this linear approximation to approximate the value of $v(t)$ at $t=2\Delta t = 6.6$.
      $v(6.6) \approx L(6.6) =$

    3. The following steps will guide you in calculating the Forward Euler approximation after the third step of size $\Delta t = 3.3$, yielding an estimate of $v(9.9)$.

      Given the value of $v(6.6)$ estimated above and the differential equation, what is the slope of $v(t)$ at time $t=2\Delta t = 6.6$?
      $\diff{ v }{t}(6.6) = 0.3 \left(- \frac{1}{170} v{\left (6.6 \right )} + 1\right) v{\left (6.6 \right )} =$

      Those two numbers are enough information to make a linear approximation at $t=2\Delta t = 6.6$.
      $L(t) = v(6.6) + \diff{ v }{t}(6.6) (t-6.6) =$

      We use this linear approximation to approximate the value of $v(t)$ at $t=3\Delta t = 9.9$.
      $v(9.9) \approx L(9.9) =$

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Math 1241, Fall 2020