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?
Best Answer
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.