Problem
Suppose q(x) and p(x) are continuous functions. Show that the set of all solutions to the following ODE gives rises to a vector space. $$y'' + py' + qy = 0$$ Let $y_1$ and $y_2$ be linearly independent solutions.
My Question
Most of my axioms for this proof rely on the multiplicative-additivity distributivity axiom but in my proof for this I use the exact result I am trying to prove. I will give the first few lines of my proof to show what I am talking about.
My Proof
Let $y_1, y_2, y_3 \in V$. We must prove that $k(y_1+y_2)=ky_1+ky_2$ for $k \in \mathbb{R}$. \begin{align*} 0&=(y_1+(y_2+y_3))''+p(y_1+(y_2+y_3))'+q(y_1+(y_2+y_3))\\[1.5ex] &=y_1''+(y_2+y_3)''+py_1'+p(y_2+y_3)'+qy_1+q(y_2+y_3) \end{align*} The expansion for '' and ' is a property of the derivative. But how do I get around using M-A distributivity in the proof for M-A distributivity. Help!