The set of all solutions to $y''+3y'+5y=0$ is a vector space.
I don't think it would be a vector space because while it does have a 0 vector, I don't think it would be closed under scalar multiplication or vector addition. Am I right in thinking so? (I'm quite new to vector space and DE both, so I'd like to make sure I understand these stuff!)
The reason the space of solutions to this homogeneous linear DE is a vector space lies in the fact that differentiation is a linear operation. Let $\phi_1,\phi_2$ be two solutions to the differential equation and let $\lambda\in\mathbb{R}$ be a scalar then: $$(\lambda \phi_1+\phi_2)''+3(\lambda \phi_1+\phi_2)'+5(\lambda \phi_1+\phi_2)=$$ $$\lambda\underbrace{(\phi_1''+3\phi_1'+5\phi_1)}_{=0}+\underbrace{\phi_2''+3\phi_2'+5\phi_2}_{=0}=\lambda0+0=0$$ and thus the set of solutions is closed under the vector space operations. It would be an affine space if the linear DE would be inhomogeneous.