Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

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 :-).

share|cite|improve this answer

Follow the method in

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.