I was asked the following question in a test.
If a Vector Space is the set of real valued continuous functions over $\mathbb R$, then find the subspace of: $$\frac{d^2y}{dx^2} - 9\frac{dy}{dx} +2y=0$$
I'm a bit confused in the question. I first thought it meant to verify for all functions which satisfy the differential equation, whether they form a subspace.
Then I thought that I should take two functions $y1$ and $y2$ and check whether $y1 + y2$ and $a.y$ where $a\in\mathbb R$ also satisfy the differential equation given that $y1$ and $y2$ satisfy it.
I'm not quite sure whether I interpreted the question correctly or whether I approached it correctly.
Can someone also post how the question should actually be worded?
Linear differential equation solutions has vector space structure. Checking it for your equation is easy. If you have two solutions $y_1,y_2$, prove $ay_1+b y_2$ is a solution. The order of your equation is the dimension of the vector space so you need to find two independet solutions of
$$ y''(x)-9y'(x)+2y(x)=0 $$
Try to resolve the equation. Solutions you should find are $y_1(x)=\exp(-(\sqrt{73}-9)x/2)$ and $y_2=\exp((\sqrt{73}+9)x/2)$.