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

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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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

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