User:Egm6321.f10.team5.abc/a4

Problem 3
Find $$X(x)$$ in terms of $$\sin$$, $$\cos$$, $$\sinh$$, and $$\cosh$$. Provided identities are $$\exp (i \theta) = \cos (\theta) + i \sin (\theta)$$ and $$\exp (\theta) = \cosh (\theta) + \sinh (\theta)$$.

Solution 3
Assume that $$X(x) = \exp(rx)$$. From the equation of $$K^4 X^{(4)} = X$$, it was already derived in the lecture notes that $$r_{1,2} = \pm \frac 1 K$$ and $$r_{3,4} = \pm \frac i K$$. Since there are four solutions for $$r$$, we make a linear combination to express $$X(x)$$. Substitute the values of $$r$$s into the equation and we get Now use the expression of exponential in terms of trigonometric functions. From the trigonometric identity, it is defined that $$\sin (- \theta) = - \sin (\theta)$$, $$\cos (- \theta) = \cos (\theta)$$, $$\sinh (- \theta) = - \sinh (\theta)$$, and $$\cosh (- \theta) = \cosh (\theta)$$. Apply the relations to the equation of $$X(x)$$. Then the equation can be simplified as