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For the PDE $u_y + u u_x = 1$, sketch a plot of $\Gamma$ and a few representative curves, including the envelope curve. Conditions: $u=0$ on the curve $y^2=2x$, and $y,x>0$. Express $u$ as a function of $x,y$.

I'm new to PDE's and very lost. I saw similar problems on here, but not identical. I'm pretty sure I have to use the Method of Characteristics. But I'm not sure how you find the curve Gamma, or parameterize everything.

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Analytic solving of the PDE : $$u_y + uu_x=1$$ Thanks to the method of characteristics : $$\frac{dy}{1}=\frac{dx}{u}=-\frac{du}{1}$$ On the characteristic curves : \begin{cases} dx+udu=0\:\:\rightarrow\:\: 2x+u^2=c_1\\ dy+du=0\:\:\rightarrow\:\: y+u=c_2\\ \end{cases} The general solution can be expressed on the form : $$\Phi\left(2x+u^2\:,\: y+u\right)=0$$ where $\Phi$ is any derivable function of two variables.

This is equivalent of the implicit equation: $$2x+u^2=F(y+u)$$ where $F$ is any derivable function.

The boundary condition $u=0$ on $2x=y^2$ implies $2x=y^2=F(y)$ Hense $F$ is the square function. $$2x+u^2=(y+u)^2$$ $$u(x,y)=\frac{2x-y^2}{2y}$$

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