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This is the pde

I am not sure how to begin this problem, I have looked up how to use the method of characteristics but can find no example where $\rho^2$ so I am unsure of how one would approach this.

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Define the characteristic curve $$ X'_a(t)=c,\;X_a(0)=a. $$ You have $X_a(t)=a+ct$. Now study the evolution of $\rho(X_a(t),t)$. You have $$ \frac{d}{dt}\rho(X_a(t),t)=\rho_t+\rho_xc=-\rho^2(X_a(t),t). $$ This ODE has an explicit solution. From here I think that you can continue :-).

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Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

$\dfrac{dt}{ds}=1$ , letting $t(0)=0$ , we have $t=s$

$\dfrac{dx}{ds}=c$ , letting $x(0)=x_0$ , we have $x=cs+x_0=ct+x_0$

$\dfrac{d\rho}{ds}=-\rho^2$ , we have $\rho(x,t)=\dfrac{1}{s+f(x_0)}=\dfrac{1}{t+f(x-ct)}$

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