I'm given(adsbygoogle = window.adsbygoogle || []).push({});

[tex]y''-4y'+4y=0[/tex]

and there is a solution [tex]y_{1}=e^{2x}[/tex]

Using this I need to find a second solution.

Starting with the assumption:

[tex]y_{2}=u(x)*y_{1}[/tex]

Then:

[tex]y=ue^{2x}[/tex]

[tex]y'=2ue^{2x}+u'e^{2x}[/tex]

[tex]y''=4ue^{2x}+2u'e^{2x}+u''e^{2x}+2u'e^{2x}[/tex]

When I substitute back into the original equation, after doing the cancellation I wind up with:

[tex]u''e^{2x}=0[/tex]

Then using the following order reduction:

[tex]w=u'[/tex]

I get:

[tex]w'e^{2x}=0[/tex]

I'm not sure what to do next here. The example in my textbook had two terms, and they used an integrating factor. How can I go about solving this problem? The answer given in the book is:

[tex]y_{2}=e^{2x}[/tex]

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# Finding A Second Solution For A DE Via Reduction Of Order

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