(a) Derive the form for the general solution to the

QUESTION:

(a) Derive the form

                           \(y(x)=A_{1} e^{x}+A_{2} e^{-x}+A_{3} \cos x+A_{4} \sin x\)

for the general solution to the equation \(y^{(4)}=y\), from the observation that the fourth roots of unity are \(1 .-1, i\), and \(-i\).
(b) Derive the form

                          \(y(x)=A_{1} e^{x}+A_{2} e^{-x / 2} \cos (\sqrt{3} x / 2)+A_{3} e^{-x / 2} \sin (\sqrt{3} x / 2)\)

for the general solution to the equation \(y^{(3)}=y\), from the observation that the cube roots of unity are \(1, e^{i 2 \pi / 3}\), and \(e^{-12 \pi / 3}\).

Equation Transcription:

Text Transcription:

y(x)=A_1e^x+A_2e^-x+A_3cos ⁡x+A_4sin ⁡x

y^(4)=y

 1,-1,i

-i

y(x)=A_1 e^x+A_2 e^-x/2 cos⁡(sqrt 3x/2) +A_3 e^-x/2 sin⁡(sqrt 3x/2)

y^(3)=y

1,e^i2pi/3

e^-12pi/3