Differential equation

Question

prove that e^ax cos(bx) and e^ax sin(bx) would be linearly independent solutions of the equation

d²y/dx² − (α1 + α2)dy/dx + α1α2y = 0

when α1 = a + bi and α2 = a − bi, where b≠0.

Answer 1

The Wronscian of y₁= e^axcosbx and y₂ = e^axsinbx

is equal to b²e^axcos2bx ≠ 0