Solve the ODE when general solution is given

Question

Given the following ODE:

        

its general solution is:

             .

Determine the value for A, then find      and     .

Please show ALL work. 

Answer 1

Characteristic equation: r⁴ + Ar² - 4 = 0.

Now from general solution we have that r = ±1 are solutions for these equarion, so 1 + A - 4 = 0, A = 3.

So, our characteristic equation is: r⁴+ 3r² - 4 = 0, (r² - 1)(r² + 4) = 0, r = ±1 and r = ±2i.

So y₃(x) = C₃cos(2x); y₄(x) = C₄sin(2x)