Given [tex]y''-4y=2[/tex](adsbygoogle = window.adsbygoogle || []).push({});

and that one solution is [tex]y_{1}=e^{-2x}[/tex]

I need to find a second solution of the homogeneous equation, and then a particular solution of the homogeneous equation.

Here's how I solved the homogeneous equation:

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

plugging into the equation:

[tex]u''e^{-2x}-2u'e^{-2x}+4ue^{-2x}-2u'e^{-2x}-4ue^{-2x}=0[/tex]

[tex]u''-4u=0[/tex]

Then using [tex]w=u', w'=u''[/tex]

[tex]w'=4w[/tex]

and I eventually got y to be [tex]y_{2}=\frac{e^{2x}/4}c[/tex]

The book says the answer is [tex]y=e^{2x}[/tex], so I'm not sure whether or not that factor of 1/4 should be there. Then for the second part of finding a particular solution, the answer is -1/2, but I'm not sure how that's arrived at either.

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# Homework Help: Solving A Non-Homogenous DE Using Reduction Order

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