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Well I have given the following PDE:

$y\cdot\partial_xu(x,y)+\partial_yu(x,y)=0$

Now I have to solve it by using the method of characteristics.

The coefficient are $(a,b,c)=(y,1,0)$. Then I have to solve the following differential equations: $\dot x(t) = y$ and $\dot y=1$. $\Longrightarrow y(t)=t+c$ and $x(t)=\frac {t^2}2+tc +d$ with any constant numbers $c,d$.

But now I don't know how to continue. How do I get $u(x,y)$ and is my ansatz correct?

Thanks for helping!!

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Yes your ansatz is correct. Now you only have to use the fact that $u$ remains constant along a characteristic which is parameterized by $(x,y) = (\tfrac12 t^2 +t c +d,t+c).$ Eliminating, $t$ we have the explicit equation $y^2-2x =c^2 -2d.$ Setting $u=f(c^2-2d)$ along the characteristic, we have $u(x,y)= f(c^2-2d)= f(y^2-2x)$ as the solution of the differential equation with $f(x)$ an arbitrary function.

Edit:

We eliminate $t$ by writing down the expression of $y^2$ in terms of $t$ and reexpress it in terms of $x$, $y^2 = (t+c)^2 = t^2 +2 t c + c^2 = 2x + c^2 -d.$

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    THANK YOU VERY MUCH! now I get everything! thx!2012-04-19