Solve the initial value problem $u_x^2u_t-1=0$, $u(x,0)=x$.
This becomes $u_x^2u_t=1$, $u(x,0)=x$.
I was thinking that this was a nonlinear wave equation at first, but the $x$ component is multiplying the $t$ component. My textbook has been going through 'let $p$ be this and $q$ be that'. Am I supposed to set $p=u_x$ and $q=u_t$? But where do I go from there?
I don't know where to start on this, let alone where to go. Any help/hints would be greatly appreciated.
$u_x^2u_t-1=0$
$u_t-\dfrac{1}{u_x^2}=0$
$u_{tx}+\dfrac{2u_{xx}}{u_x^3}=0$
Let $v=u_x$ ,
Then $v_t+\dfrac{2v_x}{v^3}=0$ with $v(x,0)=1$
Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dt}{ds}=1$ , letting $t(0)=0$ , we have $t=s$
$\dfrac{dv}{ds}=0$ , letting $v(0)=v_0$ , we have $v=v_0$
$\dfrac{dx}{ds}=\dfrac{2}{v^3}=\dfrac{2}{v_0^3}$ , letting $x(0)=f(v_0)$ , we have $x=\dfrac{2s}{v_0^3}+f(v_0)=\dfrac{2t}{v^3}+f(v)$ , i.e. $v=F\left(x-\dfrac{2t}{v^3}\right)$
$v(x,0)=1$ :
$F(x)=1$
$\therefore v=1$
$u_x=1$
$u(x,t)=x+g(t)$
$u_t=g_t(t)$
$\therefore g_t(t)=1$
$g(t)=t+C$
$\therefore u(x,t)=x+t+C$
$u(x,0)=x$ :
$C=0$
$\therefore u(x,t)=x+t$
