User:Egm6321.f11.team2.Xia/RP5.3

=R*5.3 -Expressions for $$ \displaystyle X(x)$$=

Given
Given L4-ODE-CC,

$$\displaystyle X^{(4)} - K^4 X = 0 $$  (5.3.1)

Assuming

$$\displaystyle X(x) = e^{(rx)} $$  (5.3.2)

Where

$$\displaystyle r_{1,2} = \pm K $$

$$\displaystyle r_{3,4} = \pm i \, K $$  (5.3.3)

Find
In terms of $$ \displaystyle \cos Kx, \sin Kx, \cosh Kx, \sinh kh$$

Find expression for $$ \displaystyle X(x)$$

Solution
According to the (5.3.3),

$$\displaystyle X(x) = C_1 e^{Kx} + C_2 e^{-Kx} + C_3 e^{iKx} + C_4 e^{-iKx} $$  (5.3.4)

Since

$$ \displaystyle e^{iKx} = \cos Kx +i\sin Kx$$

$$ \displaystyle e^{Kx} = \cosh Kx + \sinh Kx$$

Substituting the above relations to (5.3.4)

$$\displaystyle X(x) = C_1 (\cosh Kx \ + \sinh Kx) + C_2 (\cosh Kx \ - \sinh Kx) + C_3 (\cos Kx + i \sin Kx) + C_4 (\cos Kx - i\sin Kx) $$  (5.3.5) $$\displaystyle X(x) = c_1 \cosh Kx \ + c_2 \sinh Kx + c_3 \cos Kx  + c_4 \sin Kx $$  (5.3.6) Where

$$\displaystyle c_1 = C_1 + C_2 \in \mathbb R $$

$$\displaystyle c_2 = C_1 - C_2 \in \mathbb R $$

$$\displaystyle c_3 = C_3 + C_4 \in \mathbb R $$

$$\displaystyle c_4 = i(C_3 - C_4) \in \mathbb R$$