The differential equation, representing the curve y = eˣ(a cosx + b sinx) where a and b are arbitrary constants, is

Differentiate y = eˣ(a cosx + b sinx): dy/dx = eˣ(a cosx + b sinx) + eˣ(–a sinx + b cosx) = y + eˣ(–a sinx + b cosx). So dy/dx – y = eˣ(–a sinx + b cosx). Differentiate again: d²y/dx² – dy/dx = eˣ(–a sinx + b cosx) + eˣ(–a cosx – b sinx) = (dy/dx – y) – y. So d²y/dx² – dy/dx = dy/dx – 2y, giving d²y/dx² – 2(dy/dx) + 2y = 0.