Advanced High-School Mathematics

314 CHAPTER 5 Series and Differential Equations

3. Solve the Bernoulli ODE

(a) y′+

3

x

y=x^2 y^2 , x > 0

(b) 2y′+

1

x+ 1

y+ 2(x^2 −1)y^3 = 0

5.5.4 Euler’s method

In this final subsection we shall discuss a rather intuitive numerical
approach to solving a first-order ODE of the formy′=F(x,y), y 0 =
y(x 0 ).What we do here is to specify astep size, sayh, and proceed to
approximatey(x 1 ), y(x 2 ), y(x 3 ),...,wherex 1 =x 0 +h, x 2 =x 1 +h=
x 0 + 2h, and so on.

The idea is that we use the first-order approximation

y(x 1 )≈y(x 0 ) +y′(x 0 )(x 1 −x 0 ) =y(x 0 ) +y′(x 0 )h.

Notice thaty′(x 0 ) =F(x 0 ,y 0 ); we sety 1 =y(x 0 ) +F(x 0 ,y 0 )h, giving
the approximationy(x 1 )≈y 1. We continue:

y(x 2 ) ≈ y(x 1 ) +y′(x 1 )(x 2 −x 1 ) (first-order approximation)

≈ y 1 +y′(x 1 )h (sincey(x 1 )≈y 1 )

≈ y 1 +F(x 1 ,y 1 )h (sinceF(x 1 ,y(x 1 ))≈F(x 1 ,y 1 )).

Continuing in this fashion, we see that the approximationy(xn+1)≈
yn+1at the new pointx=xn+1is computed from the previous approx-
imationy(xn)≈ynat the pointxnvia

y(xn+1) ≈ yn+1 = yn+F(xn,yn)h.

Example 1. Approximate the solution of the ODE y′ = x+y−
1 , y(0) = 1 on the interval [0,2] using step size h = 0.2; note that
F(x,y) =x+y−1: