Solving for a Differential Equation with Inspection.

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SUMMARY

The forum discussion centers on solving a differential equation through inspection, specifically the equation involving terms with variables x, y, m, and n. The user Kiziaru expresses difficulty in manipulating the equation to isolate terms involving m and n as exponents. Participants suggest strategies such as multiplying the entire equation by a suitable factor and recognizing exact differential forms, particularly the relationship d(xy) = xdy + ydx. The conversation highlights the importance of understanding exact equations and manipulation techniques in solving differential equations.

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Kiziaru
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I'm rather bad at solving by inspection, and right now I am stuck on a problem I can't solve.1.http://www.texify.com/img/%5CLARGE%5C%21y%28x%5E2y%5E2-m%29dx%2Bx%28x%5E2y%5E2%2Bn%29dy%3D0.gif
2.The attempt at a solution
http://www.texify.com/img/%5CLARGE%5C%21x%5E2y%5E3dx-mdx%2Bx%5E3y%5E2dy%2Bndy%3D0.gif

Rearrange and divide by x^2 * y^2

http://www.texify.com/img/%5CLARGE%5C%21ydx%2Bxdy%20%3D%20%28mydx-nxdy%29/x%5E2y%5E2.gif

I know that xdy+ydx = d(xy) buy I don't know what to do with the other side. If I try to make the right side into d(mx/ny) I'm left over with an x^2 that I don't know how to get rid of.
 
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Welcome to PF!

Hi Kiziaru! Welcome to PF! :smile:

(I haven't actually tried to solve this :redface:, but …)

my immediate reaction on seeing that m and n was to think "xm and yn"
 


tiny-tim said:
Hi Kiziaru! Welcome to PF! :smile:

(I haven't actually tried to solve this :redface:, but …)

my immediate reaction on seeing that m and n was to think "xm and yn"

Thanks! I've always lurked on this site, but I never posted because most of what I needed help with was already answered, until today. :frown:

Also, I hate to sound dumb, but what wizardry is this? How can I make m and n into exponents? And is this part of a plan to get d(arctan(y/x))?
 
Kiziaru said:
How can I make m and n into exponents? And is this part of a plan to get d(arctan(y/x))?

no specific plan … just a vague idea :smile:

remember, you can multiply the whole equation by anything you like :wink:
 
What I see is that if you ignore the terms with m and n, the other part is exact...
 

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