# First Order QL-PDE using characteristic

Rather than post two questions ill ask one question involving the two.

I have to solve the First Order Quasi Linear PDE using the characteristic equation, and show the general solution satisfies the PDE:

$$x \frac{\delta U}{\delta x}+\frac{\delta U}{\delta y}=y$$ My work; $$\frac{dx}{x}=\frac{dy}{1}=\frac{du}{y}$$ Solving the first two; $$\int\frac{dx}{x}=\int{dy}$$ $$x=Ae^{y}$$ Now let; $$\int {dy}=\int\frac{du}{y}$$ $$\frac {y^2 }{2}+B=u$$ Let $B=F(A)$ $$U(x,t)=\frac {y^2 }{2}+F(xe^{-y})$$ So I think this suffices as a solution but then how do I show the general solution satisfies the PDE, find the partial derivatives and insert this into the original PDE?

So my second question involves another first order QLPDE in the form which I cant solve or at least doesn't look right: $$2y \frac{\delta U}{\delta x}+\frac{\delta U}{\delta y}=x$$ Characteristic: $$\frac{dx}{2y}=\frac{dy}{1}=\frac{du}{x}$$ Solving First two; $$\int dx=\int 2ydy$$ $$x=y^2 +C$$ Now solving the second two; $$\int dy=\int\frac{du}{x}$$ So I'm not sure what to do from here? I know I could substitute the $x$ value in but this doesn't seem right? Again I have to show the general solution satisfies the PDE. I feel like my professor hasn't discussed this in enough detail and that's why I'm struggling he wasn't much help when I went to see him either. Any guidance on the first and second question and whether I'm right or wrong appreciated.

• I have found the solution to this question, could somebody advise me should I complete the solution or close the question? What is more appropriate? – pi-e Nov 21 '15 at 13:32