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The differential equation $ xu_x + yu_y = 2u$ satisfying the initial conditions $y = xg(x), u=f(x)$ with

  1. $f(x) = 2x, g(x) = 1$, has no solution

  2. $f(x) = 2x^2, g(x) =1$, has infinite number of solutions

  3. $f(x) = x^3, g(x) = x$, has a unique solution

  4. $f(x) = x^4, g(x) = x$, has a unique solution

I don't know when a partial differential equation has unique solution, infinite solutions or no solution. can I solve this by Cauchy's method of characteristics? I have no idea. please help.

adember
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    I can't find any initial conditions. All I can see is an equation relating the independent variables $y=xg(x)$ such that they're not independent any more and an equation stating that the prospective solution $u$ is a function of $x$ only? – Max Herrmann Jul 31 '15 at 12:43

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