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?
Best Answer
Are you considering $d$ to be an arbitrary positive integer?
Then the answer is yes. In fact, one can show the following is true:
For the homogeneous equation with real coefficients: $$\tag{1}\def\sss{} a_{\sss n} y^{\sss(\!n\!)} +a_{\sss n\!-\!1} y^{\sss(\!n\!-\!1\! )} +\cdots+a_{\sss1} y' +a_{\sss0} y = 0,\quad a_n\ne0 $$
The characteristic polynomial (c.p.) is $$ \tag{2}a_{\sss n}x^n+a_{\sss n\!-\!1}x^{n-1} +\cdots+a_{\sss1}x +a_{\sss0} . $$
Find the roots and their multiplicities of (2).
If $c$ is a real root of (2) with multiplicity $k $, then $k$-independent solutions of (1) are $$ e^{ct}, xe^{ct}, x^2 e^{ct}, \ldots, x^{k-1}e^{ct} $$
If $a+bi$ is a complex root of (2) with multiplicity $k$, then $2k$-independent solutions of (1) are $$ e^{at}\sin (bt), x e^{at}\sin (bt), \ldots, x^{k-1} e^{at}\sin (bt) $$ $$ e^{at}\cos (bt), xe^{at}\cos (bt) , \ldots, x^{k-1}e^{at}\cos (bt) $$
Moreover, the set of all solutions found from the above will be independent (for instance, one can compute the Wronskian) and there will be $n$ of them (this follows from the Fundamental Theorem of Algebra).
The general solution to (1) is
$$y_c=c_{\sss 1}y_{\sss1}+c_{\sss2}y_{\sss2}+\cdots+c_{\sss n}y_{\sss n}$$
where $y_1$, $y_2$, $\ldots\,$, $y_n$ are the $n$-solutions found above.