How to find fundamental set of solutions of complementary equation of a given differential equation

Question

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

Answer 1

$$4 x^2 y'' + (4x-8 x^2) y' + (4 x^2-4x-1) y = 4 x^{1/2} e^x (1+4 x).\tag 1$$ The form of the term on the right with $e^x$ as factor draw us to a change of function $$\quad y(x)=e^xu(x)$$ in order to eliminate the exponential term.

$y'=e^x(u+u')$

$y''=e^x(u+2u'+u'')$

$$4 x^2 (u+2u'+u'') + (4x-8 x^2) (u+u') + (4 x^2-4x-1) u = 4 x^{1/2} (1+4 x).$$ After simplification :

$$4 x^2u''+ 4x u' -u = 4 x^{1/2} (1+4 x). \tag 2$$ The solution of the homogeneous equation $\quad 4 x^2u{_h}''+ 4x u{_h}' -u_h =0\quad$ is easy to find : $$u_h=c_1x^{1/2}+c_2x^{-1/2}$$ Then one have to find a particular solution of Eq.$(2)$ (In fact any one particular solution).

One can use the method of variation of parameter http://mathworld.wolfram.com/VariationofParameters.html

A particular solution of $\quad 4 x^2u''+ 4x u' -u = 4 x^{1/2}\quad$ is for example $\quad 2x^{3/2}$.

A particular solution of $\quad 4 x^2u''+ 4x u' -u = 16 x^{3/2}\quad$ is for example $\quad x^{1/2}\ln(x)$.

Thus the general solution of Eq.$(1)$ is :

$$u=c_1x^{1/2}+c_2x^{-1/2}+2 x^{3/2}+x^{1/2}\ln(x)$$ $$y=e^x\big(c_1x^{1/2}+c_2x^{-1/2}+2 x^{3/2}+x^{1/2}\ln(x)\big)$$