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Am I going about this question the right way?

Solve $u_t + x^2tu_x = 0 $ with initial condition $u_0(x) = \cos x$

I first started by finding the vector field for where $u$ is constant which is $$(x^2t,1)$$ and so I'm looking for a set of curves such that $$\frac{d}{d\tau}(x(\tau), t(\tau)) = (x(\tau)^2t(\tau), 1)$$ and so I got that $$t(\tau) = \tau$$ and so after inverting $t(\tau) = \tau$ we're looking for a function $x(t)$ such that $$\dot{x}(t) = x(t)^2t$$ but I can't think of such a function as this requires solving a non-linear ODE, am I on completely the wrong track?

EDIT

I have thought of a function! It wasn't too complicated after all, the function is $x(t) = -2t^{-2}$. I will attempt the solution now and make another edit.

EDIT 2: On second thoughts I'm missing the constant $c$, but when I make $x(t) = -2t^{-2} + c$ it makes the function really complicated when squaring etc.. does this matter?

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    In response to Edit 2, you cannot simply add the $+c$ as an afterthought -- you have to add it as soon as you do the integration. The result is $x(t)=\frac{2}{c-t^2}$.2012-05-12
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    Your I.C. is not clear. Does it means $u(x,0)=\cos x$ ?2012-06-30
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    This question has been solved perfectly. Hope that the asker has been diving enough and accept the answer at an early date.2012-09-10

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