327bcf52-b588-4962-8854-88944c2eb6cf 22 4 5 false false 0 decimal

&[DATE] &[TIME] - &[FILENAME]

3.A)

Analytic

d 2^Φ * dx 2^/ Φ L / - S D / - ≡

Φ A x L / cosh * B x L / sinh * + C + ≡

A L 2^/ x L / cosh * B L 2^/ x L / sinh * + A L 2^/ x L / cosh * B L 2^/ x L / sinh * + C L 2^/ + (- x S D / - ≡

C L 2^/ x S D / ≡

C x S Σ.a / ≡

a 2 / Φ 0 ≡

A L / x L / sinh * B L / x L / cosh * + 0 ≡

BC

0 J 0 ≡

Then

and at x=0

B 0 ≡

Φ A x L / cosh * C + ≡

A x L / cosh * C + 0 ≡

A x S Σ.a a 2 L * / cosh * / - ≡

At x=a/2

Then

Φ x S Σ.a / x L / cosh a 2 L * / cosh / * - x S Σ.a / + ≡

x Φ.analytic x S Σ.a / 1 x L / cosh a 2 L * / cosh / - (* :

Series

d 2^Φ * dx 2^/ Φ L / - x s D / - ≡

a 2 / ± Φ 0 ≡

x S S k x a 2 / + (* - exp * :

x Φ A.n x Ψ.n * n 1 ∞ sum ≡

x Φ C.m 2 m * 1 + (π * x * a / cos * m 0 ∞ sum C.n 2 n * (π * x * a / sin * n 1 ∞ sum + ≡

x s S.n x Ψ.n * n 1 ∞ sum ≡

x S S.m 2 m * 1 + (π * x * a / cos * m 0 ∞ sum S.n 2 n * (π * x * a / sin * n 1 ∞ sum + ≡

x S 2 m * 1 + (π * x * a / cos * x a 2 / - a 2 / int S.m a * 2 / ≡

x S 2 n * (π * x * a / sin * x a 2 / - a 2 / int S.n a * 2 / ≡

S.m 2 a / x S 2 m * 1 + (π * x * a / cos * x a 2 / - a 2 / int * :

S.n 2 a / x S 2 n * (π * x * a / sin * x a 2 / - a 2 / int * :

C.m S.m Σ.a 1 L 2^2 m * 1 + (π * a / (2^* + (* / :

C.n S.n Σ.a 1 L 2^2 n * (π * a / (2^* + (* / :

x Φ.series C.m 2 m * 1 + (π * x * a / cos * m 03 sum C.n 2 n * (π * x * a / sin * n 14 sum + :

S 1 :

D 1 :

L 2 :

a 20 :

k .05 :

Σ.a D L 2^/ :

x Φ.analytic x a 2 / - a 2 / explicit x Φ.series x a 2 / - a 2 / explicit

x Φ.test1 2 a / x S 20 * 1 + (π * x * a / cos * x a 2 / - a 2 / int * Σ.a 1 L 2^20 * 1 + (π * a / (2^* + (* / 20 * 1 + (π * x * a / cos * 2 a / x S 21 * (π * x * a / sin * x a 2 / - a 2 / int * Σ.a 1 L 2^21 * (π * a / (2^* + (* / 21 * (π * x * a / sin * + :

x Φ.test2 2 a / x S 21 * 1 + (π * x * a / cos * x a 2 / - a 2 / int * Σ.a 1 L 2^21 * 1 + (π * a / (2^* + (* / 21 * 1 + (π * x * a / cos * 2 a / x S 22 * (π * x * a / sin * x a 2 / - a 2 / int * Σ.a 1 L 2^22 * (π * a / (2^* + (* / 22 * (π * x * a / sin * + :

x Φ.test3 2 a / x S 22 * 1 + (π * x * a / cos * x a 2 / - a 2 / int * Σ.a 1 L 2^22 * 1 + (π * a / (2^* + (* / 22 * 1 + (π * x * a / cos * 2 a / x S 23 * (π * x * a / sin * x a 2 / - a 2 / int * Σ.a 1 L 2^23 * (π * a / (2^* + (* / 23 * (π * x * a / sin * + :

x Φ.test4 2 a / x S 23 * 1 + (π * x * a / cos * x a 2 / - a 2 / int * Σ.a 1 L 2^23 * 1 + (π * a / (2^* + (* / 23 * 1 + (π * x * a / cos * 2 a / x S 24 * (π * x * a / sin * x a 2 / - a 2 / int * Σ.a 1 L 2^24 * (π * a / (2^* + (* / 24 * (π * x * a / sin * + :

x Φ.test5 2 a / x S 24 * 1 + (π * x * a / cos * x a 2 / - a 2 / int * Σ.a 1 L 2^24 * 1 + (π * a / (2^* + (* / 24 * 1 + (π * x * a / cos * 2 a / x S 25 * (π * x * a / sin * x a 2 / - a 2 / int * Σ.a 1 L 2^25 * (π * a / (2^* + (* / 25 * (π * x * a / sin * + :

x Φ.test6 2 a / x S 25 * 1 + (π * x * a / cos * x a 2 / - a 2 / int * Σ.a 1 L 2^25 * 1 + (π * a / (2^* + (* / 25 * 1 + (π * x * a / cos * 2 a / x S 26 * (π * x * a / sin * x a 2 / - a 2 / int * Σ.a 1 L 2^26 * (π * a / (2^* + (* / 26 * (π * x * a / sin * + :

x Φ.test7 2 a / x S 26 * 1 + (π * x * a / cos * x a 2 / - a 2 / int * Σ.a 1 L 2^26 * 1 + (π * a / (2^* + (* / 26 * 1 + (π * x * a / cos * 2 a / x S 27 * (π * x * a / sin * x a 2 / - a 2 / int * Σ.a 1 L 2^27 * (π * a / (2^* + (* / 27 * (π * x * a / sin * + :

x Φ.test x Φ.test1 x Φ.test2 + x Φ.test3 + x Φ.test4 + x Φ.test5 + x Φ.test6 + x Φ.test7 + :

I.1 x Φ.analytic x a 2 / - a 2 / int : 40.0279

I.2 x Φ.test x a 2 / - a 2 / int : 40.0017

diff I.2 I.1 - abs I.1 / (100 * : 0.0655

Therefor the number of series required is 7