Problem: Find the characteristics of(adsbygoogle = window.adsbygoogle || []).push({});

[tex] xyu_x+(2y^2-x^6)u_y=0[/tex]

So I rewrote this as [tex]u_x+\frac{2y^2-x^6}{xy}u_y=0 [/tex] and then set this as [tex]

\frac{du}{dx}=0\implies \frac{dy}{dx}=\frac{2y^2-x^6}{xy} [/tex]

I solved this, and found that the characteristics were [tex]\frac{y^2+x^6}{x^4}=C[/tex]

where C is a constant, and u is constant along this curve. Now the problem says consider the initial condition [tex]u(x,α x^n)=x^2,\;\;n\in \mathbb{N}\;\;α>0,[/tex]

for what α>0 does the problem have a solution? For what α > 0 is the solution uniquely? Your answer may depend on n (Try n=1, n=2 etc.).

So I wrote [tex]αx^n=\frac{y^2+x^6}{x^4} [/tex] and solved for α, but I don't think this is what I am suppose to do, can someone help me please?

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# Method of Characteristics help

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