Third order linear differential equation

Question

castiel · Answer

$$y _{1}=-x ^{2}$$
$$y _{2}(x)=3x ^{2}lnx$$
are solutions to 
$$y \prime \prime \prime +p(x)y \prime \prime+g(x)y \prime=0   $$
$$x>0$$

Is $$y(x)=2x ^{2}-10x ^{2}lnx$$ a solution to equation?

castiel · Answer

How do I test that, I know that if y(x) is a solution then  Cy(x), C being any real number is also a solution, and if y1(x) and y2(x) are solutions then y1(x)+y2(x) is also a solution. But is there something I'm missing?

castiel · Answer

If something isn't a solution how do I show it isn't a solution?

castiel · Answer

Wait, if Cy(x) can be a solution and two solutions also give a solution then can I write

-10/3 * y2 as one solution and -2*y1 as another and adding those two we get y(x) as above. We just proved it is a solution?