- #1
member 428835
Homework Statement
$$\frac{\partial u}{\partial t} + x^2 + t+\left(\frac{\partial u}{\partial x}\right)^2 = 0\\
u(x,0)=0$$
Homework 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##.
The Attempt at a Solution
Write equation 1 as ##\ddot{x} = 2 \dot{u_x}##. Next substitute equation 3 into 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?