Given the differential equation $$4 x^2 y'' + (4x-8 x^2) y' + (4 x^2-4x-1) y = 4 x^{1/2} e^x (1+4 x).$$ I have found a general solution to a differential equation to be $$y_G = e^x ~ ( 2 x^{3/2} + x^{1/2} Ln(x) - c_1 x^{-1/2} + c_2 x^{1/2} ).$$
I am given that $~y_1 = e^x x^{1/2}~$ satisfies the complementary solution of the same differential equation. I need to find a fundamental set of solutions $~\{y_1,y_2\}~$ of the complementary equation of the original differential equation. How do I go about finding that?
One approach is to use two solutions by giving values to $~c_1~$ and $~c_2~$ and take the difference between these two solutions as another solution which becomes the second member of the fundamental set of equations or $~y_2~$. I don't have a method which consistently works using this approach.
$~y_1~$ and $~y_2~$ form a basis of a vector space. Which vector space are we talking about here? All vectors generated from $~y_1~$ and $~y_2~$?
How do I determine what $~y_2~$ is given $~y_1~$ for a given solution to a differential equation?
MM