Differential Equations 1

Question 1

What is Picard’s method of successive approximation?

Answer 1

Start with an initial guess y₀(x), then improve it.
y₀(x) = b
yₙ₊₁(x) = b + ₐ∫ˣf(t, yₙ(t)) dt

Question 2

Picard’s method of successive approximation:
Consider the differences between successive approximations. If we want yₙ to converge, what must the series of differences do?

Answer 2

e₀(x) = b
eₙ₊₁(x) = yₙ₊₁(x) - yₙ(x)
Therefore
yₙ(x) = ₀Σⁿ eₖ(x)