I’m confused, how do I answer this Eulers Method problem?

Question

Consider the solution of the differential equation y’=-2y passing through y(0)=1.

Use Eulers method with step size Δx=0.2 to estimate the solution at x=0.2, 0.4, 0.6, 0.8 and 1.0.

help!

Answer 1

You have some function y(x) that you don't know. But you do know something about the slope of the tangent line. It is defined by y' = -2y so, when y = 1, we know the slope is -2 (by just plugging in y = 1 into y' = -2y). So, now you have a point (0, 1) and a slope, (-2) and you can use those to define a line using the point-slope form:

y - y₁ = m(x - x₁)

y - 1 = -2(x - 0)

y = -2x + 1

So now we can estimate the function value at x = 0.2:

y = -2(0.2) + 1

y = 0.6

We now have a new point, (0.2, 0.6) and we can use y' = -2y to find the slope of the tangent line there. It would be y' = -2(0.6) = -1.2 and we can now create a new linear estimate of the function:

y - y₁ = m(x - x₁)

y - 0.6 = -1.2(x - 0.2)

y = -1.2x + 0.84

We can now plug in x = 0.4 to get a new estimate for y

y = -1.2(0.4) + 0.84 = 0.36

We now have a new point (0.4, 0.36) and we can use y' = -2y to find the slope of the tangent line there. etc, etc