For a differential equation like this with real coefficients: $\frac{d^2y}{dx^2}+p \frac{dy}{dx}+q y = 0$ By solving $\lambda ^2+p\lambda +q = 0$ we obtain two eigenvalue $\lambda_1 = a+bi$ and $\lambda_2 = a-bi$, if the discriminant $\Delta$ is smaller than 0. then if we want to obtain two real function solutions linearly independent,
$e^{ax}\cos(bx), e^{ax}\sin(bx)$
Then question is for a differential equation with solution space of dimension $d$, is there any guarantee that the real sub-solution space is also a space of dimension $d$. Is it always possible to find $d$ linearly independent real solutions?
I have to add that the above is just an example. What I'm asking is the general nth order ODE with n linearly independent complex solutions. Is the dimension of its real solution space the same with the complex one?