Answer
See explanation
Work Step by Step
$a)$ If $y = e^{2x}$ then $y' = 2e^{2x}$ and $y'' = 4e^{2x}$,
substitute into the equation $ y'' - 4y' + 4y = 0 $
$4e^{2x} - 4( 2e^{2x}) + 4(e^{2x})= 0$
$4e^{2x} - 8e^{2x} + 4e^{2x}= 0$
$ 0 = 0 $, Verified
If $y = xe^{2x}$ then $y' = (2x+1)e^{2x}$ and $y'' = (4x+4)e^{2x}$
substitute into the equation $ y'' - 4y' + 4y = 0 $
$(4x+4)e^{2x} - 4((2x+1)e^{2x}) + 4( xe^{2x})$
$4xe^{2x} + 4e^{2x} - 8xe^{2x} - 4e^{2x} + 4xe^{2x} = 0 $
$ 0 = 0$, Verified
$b)$ If $y = c_1e^{2x} + c_2xe^{2x}$
then $y' = 2c_1e^{2x} + c_2(2x + 1)e^{2x}$
and $y'' = 4c_1e^{2x} + c_2(4x + 4)e^{2x} $
substitute into the equation $ y'' - 4y' + 4y = 0 $
$(4c_1e^{2x} + c_2(4x + 4)e^{2x}) -4(2c_1e^{2x} + c_2(2x + 1)e^{2x}) + 4(c_1e^{2x} + c_2xe^{2x}) = 0$
$4c_1e^{2x} + 4c_1e^{2x} -8c_1e^{2x} + 4c_2xe^{2x} + 4c_2xe^{2x} -8c_2xe^{2x}+ 4c_2e^{2x} -4c_2e^{2x} = 0 $
$ 0 = 0 $, Verified
