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- Apr 13, 2013

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I want to solve the non-homogeneous equation $u_t+uu_x=0$ with the initial condition $u(x,0)=x$. Also I want to draw some of the characteristic curves.

I have tried the following so far:

The characteristic curves for $u_t+uu_x=0$ are the curves that are given by the solutions of the ode $\frac{dx}{dt}=u(x,t)$.

We have that $\frac{d}{dt}[u(x(t),t)]=u_t+uu_x=0$ and so $u(x(t),t)=c$.

We consider the line that passes through the points $(x_0,0)$ and $(x,t)$.

The slope of the line is

$\frac{x-x_0}{t-0}=\frac{dx}{dt}=u(x,t)=u(x_0,0)=x_0$, thus $x-x_0=tx_0 \Rightarrow x_0=\frac{x}{t+1}$.

Then we would get that $u(x,t)=\frac{x}{t+1}$. But this cannot be right, since $u$ should be only a function of $x$.

So have I done something wrong?