Lecture Notes in Differential Equations - Bruce E. Shapiro

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Figure 33.1: Illustration of Euler’s Method. A tangent line with slope
f(t 0 , y 0 ) is constructed from (t 0 , y 0 ) forward a distance h = t 1 −t 0 in the t−
direction to determined y 1 . Then a line with slope f(t 1 , y 1 ) is constructed
forward from (t 1 , y 1 ) to determine y 2 , and so forth. Only the first line is
tangent to the actual solution; the subsequent lines are only approximately
tangent.
y 2
y 1
y 0
t 0 t 1 t 2
Since y ′ (t) = f(t, y), we can approximate the left hand side of (33.10) by
y ′ n(t n ) ≈ f(t n , y n ) (33.11)
and hence
y n+1 = y n + h n f(t n , y n ) (33.12)
It is often the case that we use a fixed step size h = t j+1 − t j , in which case
we have
t j = t 0 + jh (33.13)
In this case the Forward Euler’s method becomes
y n+1 = y n + hf(t n , y n ) (33.14)
The Forward Euler’s method is sometimes just called Euler’s Method.
An alternate derivation of equation (33.12) is to expand the solution y(t)
in a Taylor Series about the point t = t n :
y(t n+1 ) = y(t n + h n ) = y(t n ) + h n y ′ (t n ) + h2 n
2 y′′ (t n ) + · · · (33.15)
= y(t n ) + h n f(t n , y( n )) + · · · (33.16)