Question

Let y(t) be the solution to dy/dt = -y satisfying y(0) = 2. Use Euler's method with time step h = 0.2 to approximate y(0.2), y(0.4), ..., y(1.0).

Answer 1

Euler's method is used to approximate the values of a function given its derivative and an initial condition. Starting with y(0) = 2 and a time step h = 0.2, successive values of y are calculated iteratively.

Step-by-step explanation:

The student is asked to use Euler's method to approximate the values of a function y(t) at certain points given its derivative dy/dt and an initial condition. With a time step of h = 0.2, we'll calculate y(0.2), y(0.4), ..., y(1.0) iteratively starting from y(0) = 2.

Let's perform the first step together:

• At t=0, y(0)=2, so using dy/dt = -y, we have dy = -2 * 0.2 = -0.4.
• Therefore, y(0.2) ≈ y(0) + dy = 2 - 0.4 = 1.6.

This process is repeated, using the new value of y each time to compute the next one:

1. y(0.4) is found by using y(0.2) in the same manner, and so on until y(1.0) is computed.