General solution of the ODE

Question

Find the general solution of the following ODE:

       

Please show ALL work.

Answer 1

Characteristic equation here is: r⁴ + 8r² + 16 = 0; (r² + 4)² = 0, r² + 4 = 0, r² = - 4, r = ±2i.

So we have 4 imaginary zeros 2i, 2i, - 2i - 2i.

Because general solution for this equation: y = (C₁x + C₂)cos(2x) + (C₃x + C₄)sin(2x).

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