Find the general solution of the ODE

Question

Find the general solution of the following ODE:

     

Please show ALL work. 

Answer 1

Question: Find the general solution of the following ODE:     

Answer: This is a fourth-order homogeneous linear ODE with constant coefficients. Find the characteristic polynomial and find its roots. Then use those roots to get the general solution via the superposition principle.

16m⁴ - 81 = 0

(4m² - 9)(4m² + 9) = 0

(2m - 3)(2m + 3)(2m - 3i)(2m + 3i) = 0

m = 3/2, m= -3/2, m= 3i/2, m = -3i/2

(two distinct real roots, two distinct complex roots (where a complex number takes the form a±bi, so a=0 and b=(3/2) here))

h(y) = c₁e^(3/2)y + c₂e^(-3/2)y + e^0y[c₃cos((3/2)y) + c₄sin((3/2)y)]

h(y) = c₁e^(3/2)y + c₂e^(-3/2)y + c₃cos((3/2)y) + c₄sin((3/2)y)

Answer 2

Characteristic equation for this equation: 16r⁴ - 81 = 0. (4r² - 9)(4r² + 9) = 0;

4r² - 9 = 0, r² = 9/4, r = ±3/2 or 4r² + 9 = 0, r = ±3/2i.

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

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