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_N3WTON_
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Homework Statement
Use the reduction of order metho to find a second linearly independent solution. What is the general solution of the differential equation?
[itex] y'' - y = 0 [/itex]
[itex] y_1(x) = e^x [/itex]
Homework Equations
Reduction of order formula
The Attempt at a Solution
First, I set:
[itex] y = ve^x [/itex]
[itex] y' = ve^x + e^{x}v' [/itex]
[itex] y'' = ve^x + 2e^{x}v' + e^{x}v'' [/itex]
[itex] (ve^x + 2e^{x}v' + e^{x}v'') - (ve^x) = 0 [/itex]
[itex] 2e^{x}v' + e^{x}v'' = 0 [/itex]
[itex] e^{x}v'' + 2e^{x}v' = 0 [/itex]
[itex] v'' + e^{x}v' = 0 [/itex]
Then I made a substitution:
[itex] w = v' [/itex]
So the equation becomes:
[itex] w' + e^{x}w = 0 [/itex]
At this point, I tried to find an integration factor. However, the integrating factor I obtained is a bit unusual, which leads me to believe that I have made a mistake somewhere. This is the integrating factor I obtained:
[itex] p(x) = e^x [/itex]
[itex] u(x) = e^{\int e^x} = e^{e^x} [/itex]
At this point, due to the odd integrating factor, I am not sure what I have done wrong or how to continue the problem.