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