Question

C Consider a differential equation with the given slope field and the in y(0) = 1. 0.5 st -0.5 (a) Explain why, if you wanted to approximate y(2) using two steps of Euler's method, you would need At = 1. (b) Use a straight edge to graph two steps of Euler's method to approximate y(2). (c) This time, instead of using two steps of Euler's method, sketch on the same slope field what it would look like if you used four steps of Euler's method to approximate y(2). (d) Besides the obvious difference that the step size is different, state two other things that are different between your answers to parts 2b and 2b. (e) Besides the obvious fact that they both use Euler's method, what is similar about the first step to your answers to parts 2b and 2d?

Answer 1

a)
The initial value given is y(0) = li.e. to = 0 for y = 1.But, we need to approximate y(2) using two steps i.e. t2 =
  
Δt = t2 – to 2

b) and c)

• 2 • • • • •0. Jo 2 2 2 2 2 2 150 («€'s» -Фут тау, и - 1 541 фрее-д. 2- \ \ т т т 5-6 1 т т т т т 56 1. т т т т .

d) Two Differences:

• In graph 2b, the value first increases then decreases. In graph 2c, the value gradually increases.
• In graph 2b, there are many changes or deviations from the slope field. In graph 2c, it almost follows the slope field.

e) Similarity: The both graph starts with the same initial value i.e. y(0)=1. and the initial direction f(0,1).