Given that f(x) = e^x is a solution to y''-2y' +y = 0, determine a second linearly independent solution.

Let y= v(x)f(x) = ve^x

==>
y' = ve^x + v'e^x

==> y'' = ve^x + 2v'e^x +
v''e^x

Now we will substitute into the
equation.

==> ve^x + 2v'e^x + v''e^x - 2(ve^x+v'e^x) + ve^x =
0

Since we need a second linearly independant solution, then we know
that v'' = 0

==> v= c1x +
c2

Then the second solution is given by
:

==> y= vf =
xe^x