Exercises

1. Use Euler's Method to generate approximate values \(y_1\), \(y_2\), and \(y_3\) of the solution \(y(t)\) of the initial value problem

   \(\frac{dy}{dt}=y^2\)  with  \(y(0)=1\)

   and with the step size indicated in each part.

   a.   \(\Delta t = 0.1\)

   b.   \(\Delta t = 0.01\)

   c.   \(\Delta t = 0.001\)

2. Use Euler's Method to generate approximate values \(y_1\), \(y_2\), and \(y_3\) of the \(y(t)\) solution of the initial value problem

   \(\frac{dy}{dt}=y^2+1\)  with  \(y(0)=2\)

   and with the step size indicated in each part.

   a.   \(\Delta t = 0.1\)

   b.   \(\Delta t = 0.01\)

   c.   \(\Delta t = 0.001\)

3. Use Euler's Method to generate approximate values \(y_1\), \(y_2\), and \(y_3\) of the solution \(y(t)\) of the initial value problem

   \(\frac{dy}{dt}=t^2\)  with  \(y(0)=1\)

   and with the step size indicated in each part.

   a.   \(\Delta t = 0.1\)

   b.   \(\Delta t = 0.01\)

   c.   \(\Delta t = 0.001\)

4. Find the exact solution of the initial value problem

   \(\frac{dy}{dt}=t^2\)  with  \(y(0)=1\).

   [Hint: Determine all the functions that have derivative \(t^2\), and let \(y(t)\) be the one that satisfies the initial condition.]
5. Use Euler's Method to generate approximate values \(y_1\), \(y_2\), and \(y_3\) of the solution \(y(t)\) of the initial value problem

   \(\frac{dy}{dt}=t^2+1\)  with  \(y(0)=2\)

   and with the step size indicated in each part.

   a.   \(\Delta t = 0.1\)

   b.   \(\Delta t = 0.01\)

   c.   \(\Delta t = 0.001\)

6. Find the exact solution of the initial value problem

   \(\frac{dy}{dt}=t^2+1\)  with  \(y(0)=2\).

   [Hint: Determine all the functions that have derivative \(t^2+1\), and let \(y(t)\) be the one that satisfies the initial condition.]