Page 383 - Schaum's Outline of Differential Equations

366 ANSWERS TO SUPPLEMENTARY PROBLEMS

CHAPTER 20

20.15. y' = z,z' = - y, y(G) = 1, z(0) = 0

20.16. y' = i, i' = y + x; y(0) = 0, z(0) = - 1

20.19. Method: EULER'S METHOD
Problem: /' + y = 0; y(0) = 1 , y'(0) = 0

h = 0.l
x n
True solution
y n ^n Y(x) = cos x
0.0 1.0000 0.0000 1.0000
0.1 1.0000 -0.1000 0.9950

0.2 0.9900 -0.2000 0.9801
0.3 0.9700 -0.2990 0.9553

0.4 0.9401 -0.3960 0.9211
0.5 0.9005 -0.4900 0.8776

0.6 0.8515 -0.5801 0.8253
0.7 0.7935 -0.6652 0.7648

0.8 0.7270 -0.7446 0.6967
0.9 0.6525 -0.8173 0.6216

1.0 0.5708 -0.8825 0.5403

20.20. Since the true solution Y(x) = -x, a first-degree polynomial, Euler's method is exact and generates the true solution
y n=-x nateachx n.