Solve PDE $\rho_t + c\rho_x +\rho^2 = 0$ using method of characteristics

Question

$$ \frac{\partial\rho}{\partial t} + c\frac{\partial\rho}{\partial x} + \rho^2 = 0 $$

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.

Answer 1

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 :-).

Answer 2

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 $\rho(x,t)=\dfrac{1}{s+f(x_0)}=\dfrac{1}{t+f(x-ct)}$