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I am currently trying to solve a PDE but am having difficulties. The PDE is: $$(y+u)u_x+yu_y=x-y, \>\>\>\>u(x,1)=1+x$$ Now I found the Characteristic Equations: $$\dot x(s) = y+z, \>\>\dot y(s)=y, \>\> \dot z(s)=x-y$$ Where $z(0)=u(x_0,y_0), \>\> x_0=x(0) \text{ and } y_0=y(0)$. The dot above is differentiation with respect to $s$. Then, I firstly solved for $y$ to get: $$y(s)=e^s, \>\>y(0)=1$$ Thus we have our Auxiliary data confirmed. Now, This is where I get confused. If I sub in $y$ into the $z$ equation, this gives: $$\dot z(s)=x-e^s$$Now I considered deriving this again, giving: $$\ddot z(s) = \dot x(s)-e^s = e^s+z-e^s=z$$ Am I allowed to do this? I'm not sure if this is allowed, but if so then I can continue, solving for $z(s)$ and complete the question. Thanks!

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