Eliminating First Derivative in Tricky Differential Equation | Homework Help

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SUMMARY

The discussion centers on solving the differential equation xy'' + 2y' + (n^2)x*y = sin(omega*x) by eliminating the first derivative term. Participants suggest substitutions such as u = xy' and u = xy to simplify the equation. The hint provided emphasizes recognizing the structure of the equation, particularly the resemblance of xy'' + y' to the derivative of a product. Effective manipulation of these substitutions is crucial for progressing towards a solution.

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Homework Statement


xy''+2y'+(n^2)*x*y=sin(omega*x)

Hint: Eliminate the first derivative term


Homework Equations





The Attempt at a Solution



I have tried lots of substitutions, but none of them seems to work out. I don't really understand what the hint is getting at. For example, I've tried y'=ux, but then I don't know what to do with the y. I've tried y=ux, but that doesn't seem to help matters at all. And I've tried numerous other silly things that haven't worked out. I'm really out of ideas and stuck. Any little hints would be greatly appreciated.
 
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You notice that xy'' + y' looks very much like the derivative of a product.

So I suggest you try playing around with u = xy' or x(y'2); and if that doesn't work maybe u = xy
 

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