I think I've solved this problem, but just need someone to check my answers.(adsbygoogle = window.adsbygoogle || []).push({});

[tex]u_{x}u+u_{t}=2[/tex]

With initial data

[tex]u(x,0)=f(x)[/tex]

Use the method of charactersitics [tex]u(x,t)=u(\xi,\tau)[/tex], we get

[tex]\tau_{t}=t[/tex], [tex]x_{u}=u[/tex] and [tex]u_{\tau}=2[/tex].

So, by choice of our initial condition,

[tex]\tau=t[/tex] and [tex]u=2t + f(\xi)[/tex].

Since [tex]\xi=\xi(x,t)[/tex]. We have

[tex]u(x,t)=u(\xi(x,t),t)[/tex]

At [tex]t=0[/tex] we have [tex]u(x,0)=u(\xi(x,0),0)[/tex], therefore

[tex]x=\xi(x,0)[/tex]

Since we do not know what [tex]f(x)[/tex] is, our solution for [tex]u(x,t)[/tex] is just

[tex]u(x,t)=2t+f(\xi(x,t))[/tex]

with initial condition

[tex]\xi(x,0)=x[/tex]

Is this right??

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# Homework Help: Answer need checking!

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