Appendix: All Solutions to the Problems
881
or
910a 0 + 284a 1 = 175
284a 0 + 159a 1 = 70
Solve for a 0 and a 1
a 0 =
7945
64034
= 0.12407471, a 1 =
14000
64034
= 0.21863385
Thus, the approximate solution can be obtained
y 1 = (x − 1)
2
7945
64034
+
14000
64034
x
= (x − 1)
2
(0.12407471 + 0.21863385x)
8.4 Find the approximate solution of extremal problem of the functional J [y] =
1
0 (y
2
− y
2
− 2x y)dx, and do a comparison with the accurate solution, the
boundary conditions are y(0) = 0, y(1) = 0.
Hint: The approximate solution is searched in the form of y n (x) = x(1 − x)(a 0 +
a 1 x + · · · + a n x
n
), and calculating the values of n = 0 and n = 1.
Solution Let F = y
2
− y
2
− 2x y, the Euler equation of the functional is
−2y − 2x − 2y
= 0
or
y
+ y = −x
The general solution of the homogeneous equation is
Y = c 1 cos x + c 2 sin x
Let the particular solution be
y
∗
= b 0 + b 1 x
Substituting the above expression into the Euler equation, solve for b 0 = 0,
b 1 = −1. So the general solution is
y = Y + y
∗
= c 1 cos x + c 2 sin x − x
Using the boundary conditions y(0) = 0, y(1) = 0, we get c 1 = 0, c 2 =
1
sin 1
,
thus the accurate solution is y =
sin x
sin 1
− x.
Let y n (x) = x(1 − x)(a 0 + a 1 x + · · · + a n x
n
), there are