General solution of the following ODE

Question

Finf the general solution of the following ODE:

16y⁽⁴⁾-81y=0

Please show ALL work 

Answer 1

Characteristic equation is 16r⁴ - 81 = 0; r⁴ - (3/2)⁴ = 0; (r² - 9/4)(r² + 9/4) = 0,

r = ±3/2, r = ±3/2i.

So, general solution y = C₁e^3/2x + C₂e^-3/2x + C₃cos(3/2x) + C₄sin(3/2x).

Here C₁, C₂, C₃, C₄ are arbitrary constants.