Can a differential equation with real coefficients have solution with complex coefficients?

Question

Can a differential equation (with constant coefficients, linear or nonlinear) with real coefficients have solution(s) with complex coefficients? If so, are there any examples related to actual physical systems? How about the opposite case (complex coefficient differential equation with real coefficient solution)?

Answer 1

consider $x''+4\pi^2x=0$it has two independent complex solution $e^{2\pi ix},e^{-2\pi ix}$. if f(x) is a complex solution of a real equation if and only if Re(f),Im(f) is real solutions of that equation.conversely a real function f(x) is a solution of a complex equation iff f is a solution of real and imaginary part of equation

Answer 2

Yes, imagine $$ay'' + by' +cy = 0$$ where the coefficents satisfy $$b^2 -4ac < 0$$ But usually you implicitly use euler identity $$ \mathrm{e}^{ir x} = \cos(rx) + i\sin(rx) $$ After which the complex nature is hidden in the constants of integration. I.e $$ C\mathrm{e}^{ir x} = C\cos(rx) +C_1\sin(rx) $$

In physics there are plenty of areas where such equationd exist for example the damped linear spring or pendulum where you have simple harmonic motion with retardation or even just without but still satisfying the above.