Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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|improve this question

2 Answers 2

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|improve this answer

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

share|improve this answer

Your Answer

 
discard

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.