Linear Algebra - Solutions and Basis

Question

I am faced with this question. However, I do not know how to solve. I think the answer is the second option though. Anyone has any tips? Thanks in advance.

Given that $y'=Ay+h$ is a $2 \times 2$ non-homogeneous SDE. Suppose $y_p$ is a particular solution of the system and $x_1$ and $x_2$ is a basis for the solution space of the associated system $y' = Ay$. Which of the following is true:

1. A general solution of $y' = Ay+h$ is given by $c_1x_1+c_2x_2 + c_3y_p$.
2. The difference $x_a-x_b$ of two solutions $x_a$ and $x_b$ of $y'=Ay+h$ is a solution of $y' = Ay$.
3. $x_1, x_2$ and $y_p$ is a basis for the solution space of $y' = Ay+h$.
4. $x_1+y_p$ and $x_2+y_p$ is a basis for the solution space of $y' = Ay+h$.
5. None of the above.

Answer 1

HINTS

1. Clearly, $c_1x_1+c_2x_2 + y_p$ is a solution. Is $c_1x_1+c_2x_2 + 2y_p$ a solution? Even simpler, is $2y_p$ a solution?

2. So you have $x_a' = Ax_a+h$ and $x_b' = Ax_b+h$. Is it true that $(x_a-x_b)' = A(x_a-x_b)$?

3. Are the solutions to $y'=Ay+h$ even a vector space? In particular, is $0$ a solution? If not, how can we talk about a span of something that is not a vector space?