TITLE: Solutions of homogeneous linear differential equation form a vector space
QUESTION [5 upvotes]: Show that the solutions of a homogeneous linear differential equation $y''+a(x)y'+b(x)y = 0$ form a vector space. What is its dimension?

I understand that the dimension is 2 and that 0 is a solution to the differential equation ($0''+a(x)*0'+b(x)*0=0$).
How does one go about proving the other two properties of a vector space: closed under addition and closed under multiplication?

REPLY [9 votes]: Let $y_1,y_2$ be solutions for the equation
$$y''+ay'+by=0.$$
Then
$$y_1''+ay_1'+by_1=0$$
$$y_2''+ay_2'+by_2=0$$
Add the first and the second equation we have
$$(y_1+y_2)''+a(y_1+y_2)'+b(y_1+y_2)=0.$$
Then $y_1+y_2$ is solution too.
Let $\lambda$ be a real number
$$\lambda y_1''+a\lambda y_1'+b\lambda y_1=\lambda(y_1''+ay_1'+by_1)=0.$$
Then $\lambda y_1$ is solution too.