The Computable Differential Equation Lecture ... - Bruce E. Shapiro

56 CHAPTER 3. APPROXIMATE SOLUTIONS
Figure 3.3: Geometry for Theorem 3.8
3.5 Euler’s Method for Systems
The numerical methods we will discuss in this class can all easily be extended to
systems. For a system, we have
y ′ = f(t, y) (3.115)
y(t 0 ) = y 0 (3.116)
To emphasize that y and y 0 are vectors and that f is a vector function we have
written them all in bold-face. Euler’s method then becomes
Example 3.2. Solve the initial value problem
on the interval [0, π] using h = π/4.
y n+1 = y n + h n f(t n , y n ) (3.117)
y ′ 1(t) = y 2 (t) (3.118)
y ′ 2(t) = −2y 1 (t) (3.119)
y 1 (0) = 1 (3.120)
y 2 (0) = 0 (3.121)
Solution. Let y = (y 1 , y 2 ) T . Then y(0) = y 0 = (1, 0) T , and we write
f(t, y) = (y 2 , −2y 1 ) T (3.122)
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007