Math Insight

Forward Euler and linear approximations

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  1. For the following logistic differential equation \begin{align*} \diff{ x }{ t } &= 1.3 x \left(- \frac{x}{370} + 1\right)\\ x(0) & = 481 \end{align*} calculate a Forward Euler approximation to $x(2.4)$ using a time step of $\Delta t = 0.8$. 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{ x }{ t }(t) &= 1.3 \left(- \frac{1}{370} x{\left (t \right )} + 1\right) x{\left (t \right )}. \end{align*}

    1. The following steps will guide you in calculating the Forward Euler approximation after one step of size $\Delta t = 0.8$, yielding an estimate of $x(0.8)$.

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

      Based on that result and the differential equation, what is the slope of $x(t)$ at time $t=0$?
      $\diff{ x }{t}(0) = 1.3 \left(- \frac{1}{370} x{\left (0 \right )} + 1\right) x{\left (0 \right )} =$

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

      We use this linear approximation to approximate the value of $x(t)$ at $t=\Delta t = 0.8$.
      $x(0.8) \approx L(0.8) =$

    2. The following steps will guide you in calculating the Forward Euler approximation after the second step of size $\Delta t = 0.8$, yielding an estimate of $x(1.6)$.

      Given the value of $x(0.8)$ estimated above and the differential equation, what is the slope of $x(t)$ at time $t=\Delta t = 0.8$?
      $\diff{ x }{t}(0.8) = 1.3 \left(- \frac{1}{370} x{\left (0.8 \right )} + 1\right) x{\left (0.8 \right )} =$

      Those two numbers are enough information to make a linear approximation at $t=\Delta t = 0.8$.
      $L(t) = x(0.8) + \diff{ x }{t}(0.8) (t-0.8) =$

      We use this linear approximation to approximate the value of $x(t)$ at $t=2\Delta t = 1.6$.
      $x(1.6) \approx L(1.6) =$

    3. The following steps will guide you in calculating the Forward Euler approximation after the third step of size $\Delta t = 0.8$, yielding an estimate of $x(2.4)$.

      Given the value of $x(1.6)$ estimated above and the differential equation, what is the slope of $x(t)$ at time $t=2\Delta t = 1.6$?
      $\diff{ x }{t}(1.6) = 1.3 \left(- \frac{1}{370} x{\left (1.6 \right )} + 1\right) x{\left (1.6 \right )} =$

      Those two numbers are enough information to make a linear approximation at $t=2\Delta t = 1.6$.
      $L(t) = x(1.6) + \diff{ x }{t}(1.6) (t-1.6) =$

      We use this linear approximation to approximate the value of $x(t)$ at $t=3\Delta t = 2.4$.
      $x(2.4) \approx L(2.4) =$

  2. For the following differential equation \begin{align*} \diff{ z }{ t } &= - 1.8 z - 8.1\\ z(0) & = -9.1 \end{align*} calculate a Forward Euler approximation to $z(1.2)$ using a time step of $\Delta t = 0.4$. 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{ z }{ t }(t) &= - 1.8 z{\left (t \right )} - 8.1. \end{align*}

    1. The following steps will guide you in calculating the Forward Euler approximation after one step of size $\Delta t = 0.4$, yielding an estimate of $z(0.4)$.

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

      Based on that result and the differential equation, what is the slope of $z(t)$ at time $t=0$?
      $\diff{ z }{t}(0) = - 1.8 z{\left (0 \right )} - 8.1 =$

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

      We use this linear approximation to approximate the value of $z(t)$ at $t=\Delta t = 0.4$.
      $z(0.4) \approx L(0.4) =$

    2. The following steps will guide you in calculating the Forward Euler approximation after the second step of size $\Delta t = 0.4$, yielding an estimate of $z(0.8)$.

      Given the value of $z(0.4)$ estimated above and the differential equation, what is the slope of $z(t)$ at time $t=\Delta t = 0.4$?
      $\diff{ z }{t}(0.4) = - 1.8 z{\left (0.4 \right )} - 8.1 =$

      Those two numbers are enough information to make a linear approximation at $t=\Delta t = 0.4$.
      $L(t) = z(0.4) + \diff{ z }{t}(0.4) (t-0.4) =$

      We use this linear approximation to approximate the value of $z(t)$ at $t=2\Delta t = 0.8$.
      $z(0.8) \approx L(0.8) =$

    3. The following steps will guide you in calculating the Forward Euler approximation after the third step of size $\Delta t = 0.4$, yielding an estimate of $z(1.2)$.

      Given the value of $z(0.8)$ estimated above and the differential equation, what is the slope of $z(t)$ at time $t=2\Delta t = 0.8$?
      $\diff{ z }{t}(0.8) = - 1.8 z{\left (0.8 \right )} - 8.1 =$

      Those two numbers are enough information to make a linear approximation at $t=2\Delta t = 0.8$.
      $L(t) = z(0.8) + \diff{ z }{t}(0.8) (t-0.8) =$

      We use this linear approximation to approximate the value of $z(t)$ at $t=3\Delta t = 1.2$.
      $z(1.2) \approx L(1.2) =$

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Math 1241, Fall 2020