Can this method be used with variable coefficients?

Thanks.

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- Jan 2nd 2011, 05:36 PMdwsmithx^{2}u_{x}-u_{y}+u=0

Can this method be used with variable coefficients?

Thanks. - Jan 2nd 2011, 07:59 PMJester
You need to learn about the method of charactistics. Not everything admits a separation of variables.

- Jan 2nd 2011, 08:14 PMSudharaka
Dear dwsmith,

No. Suppose we have a differential equation such as, We take the trail solution as,

Why do we take this exponential function as our trail solution? This is based on the realization that the exponential function keeps its original shape even after differentiation. That is even after taking the derivative you get a the same exponential function with a different coefficient. This is necessary to substract the two functions, . The same arguement is valid for any constant coefficient linear differential equation(either partial or ordinary). But when the equation is, taking the solution as is not justifiable since the equation is not a constant coefficient one. For more information refer, Linear differential equation - Wikipedia, the free encyclopedia

Hope this helps to clarify matters. - Jan 2nd 2011, 08:26 PMdwsmith
- Jan 2nd 2011, 08:32 PMdwsmith
- Jan 3rd 2011, 08:18 PMdwsmith

Can I have an x in my m solution?

- Jan 4th 2011, 06:46 AMJester
Nope. To solve this separate variables

and integrate. - Jan 4th 2011, 10:04 AMdwsmith

Correct? - Jan 4th 2011, 03:50 PMSudharaka
- Jan 4th 2011, 03:57 PMdwsmith
I have it written down on paper correctly though :)

- Jan 4th 2011, 03:58 PMSudharaka