Problem Sheet 1#

  1. Show that the following functions satisfy the Lipschitz condition on \(y\) on the indicated set \(D\):

    a. \(f(t,y)=ty^3,\) \(D=\{(t,y);-1\leq t \leq 1, 0\leq y \leq 10\};\)

    b. \(f(t,y)=\frac{t^2y^2}{1+t^2},\) \(D=\{(t,y);0\leq t, -10\leq y \leq 10 \}.\)

  2. Apply Euler’s Method to approximate the solution of the given initial value problems using the indicated number of time steps. Compare the approximate solution with the given exact solution, and compare the actual error with the theoretical error

    a. \(y'=t-y, \ \ (0\leq t \leq 4)\) with the initial condition \(y(0)=1\), \(N=4\), with the exact solution \(y(t)=2e^{-t}+t-1.\) The Lipschitz constant is determined on \(D=\{(t,y);0\leq t \leq 4, y\in \rm I\!R \}.\)

    b. \(y'=y-t, \ \ (0\leq t \leq 2) \) with the initial condition \(y(0)=2,\) \(N=4\), with the exact solution \(y(t)=e^{t}+t+1\). The Lipschitz constant is determined on \( D=\{(t,y);0\leq t \leq 2, y\in \rm I\!R \}. \)