# Method of Characteristics

1. Mar 22, 2017

### joshmccraney

1. The problem statement, all variables and given/known data
$$\frac{\partial u}{\partial t} + x^2 + t+\left(\frac{\partial u}{\partial x}\right)^2 = 0\\ u(x,0)=0$$

2. Relevant equations
$$\dot{x} = 2 u_x ;\,\,t=0,\,\,x=\xi\\ \dot{u}=2(u_x)^2+u_t ;\,\,t=0,\,\,u=0\\ \dot{u_x}=-2x ;\,\,t=0,\,\,p=0\\ \dot{u_t}=-1 ;\,\,t=0,\,\,u_t=-\xi^2.$$
where $\dot{f}$ is the total derivative of $f$ with respect to $t$, or $\dot{f} \equiv \frac{df}{dt}$ where $x$ is a function of $t$.
3. The attempt at a solution
Write equation 1 as $\ddot{x} = 2 \dot{u_x}$. Next substitute equation 3 in to arrive at $$\ddot{x}=-4x^2 \implies\\ x = A \sin(2t) + B\cos(2t)$$ The first BC associated with equation 1 implies $B = \xi$, but now I'm stuck. Any ideas how to proceed?

2. Mar 28, 2017

### PF_Help_Bot

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