Burgers Equation Question - cannot satisfy initial conditions

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Burgers Equation Question -- cannot satisfy initial conditions

Homework Statement

Use characteristics to solve $$u_t+uu_x=0$$ on half line x≥0 with $$u(x,0)=x^2$$

Homework Equations

NA

The Attempt at a Solution

I think I have an issue with the initial condition. So solving via characteristics gives:

$$\frac{dx}{dt}=u \Rightarrow x=ut+x_0 \Rightarrow u=f(x-ut)$$. Then we plug in initial values and get:

$$u(x,0)=f(x)=x^2 \Rightarrow u=(x-ut)^2 \Rightarrow u^2t^2-u(1+2xt)+x^2=0$$

Then by quadratic formula

$$u=\dfrac{(1+2xt) \pm \sqrt{(1+2xt)^2+4x^2t^2}}{2t^2}$$

Now here I have a problem. So, in order to pic which sign solution to use, I need to use the initial condition. However, at t=0, I have an obvious problem. If anyone could offer help, or whether there is a typo in the problem, that would be much appreciated!

 

[haruspex]

at t=0, I have an obvious problem.

What is the limit as t→0+?

 

[zapz]

What is the limit as t→0+?

Thank you for a quick reply! I just did the limit out and that resolved my issue. Thank you very much!