Consider the wave equation
$${\partial^2 u\over \partial x^2}={\partial^2 u\over \partial t^2}$$
for a smooth function $u(x,t)$.
i. Letting $v={\partial u / \partial x}$ and $w={\partial u / \partial t}$, write down a system of four first-order partial differential equations for $u,v$ and $w$ which is equivalent to the wave equation.
My solution is
$${\partial v \over \partial x}={\partial^2 u\over \partial x^2}={\partial^2 u\over \partial t^2}={\partial w \over \partial t}$$
$${\partial v \over \partial t}={\partial \over \partial t}{\partial u\over \partial x}={\partial \over \partial x}{\partial u\over \partial t}={\partial w \over \partial x}$$
Is this the whole solution? The question says there are four equations in the system.
ii. Let $$G(x,t)=g''(x-a(t)) ,$$ where $g$ and $a$ are smooth. Combine the four equations from the previous part i. with the following two equations,
$${\partial v \over \partial x}+{\partial w \over \partial t}=2G(x,t)\,,\quad{\partial v \over \partial t}+{\partial w \over \partial x}=-2\dot a(t)G(x,t)\,,$$
to obtain a first-order system of the form
$${\partial u \over \partial x}=f_1^1(x,t,u,v,w) \quad {\partial u \over \partial t}=f_2^1(x,t,u,v,w)$$ $${\partial v \over \partial x}=f_1^2(x,t,u,v,w) \quad {\partial v \over \partial t}=f_2^2(x,t,u,v,w)$$ $${\partial w \over \partial x}=f_1^3(x,t,u,v,w) \quad {\partial w \over \partial t}=f_2^3(x,t,u,v,w)$$
where $f_i^\alpha(x,t,u,v,w)$ are functions that you should find explicitly.
My solution is
$${\partial v \over \partial x}={\partial w \over \partial t}\,,\quad{\partial v \over \partial t}={\partial w \over \partial x}\,,\quad{\partial v \over \partial x}+{\partial w \over \partial t}=2G(x,t)\,,\quad{\partial v \over \partial t}+{\partial w \over \partial x}=-2\dot a(t)G(x,t)\,,$$
$${\partial v \over \partial x}={\partial w \over \partial t}=G(x,t)\,,\quad{\partial v \over \partial t}={\partial w \over \partial x}=-\dot a(t)G(x,t)\,,$$
Then the system can be written as $${\partial u \over \partial x}=v \quad {\partial u \over \partial t}=w$$ $${\partial v \over \partial x}=G(x,t) \quad {\partial v \over \partial t}=-\dot a(t)G(x,t)$$ $${\partial w \over \partial x}=-\dot a(t)G(x,t) \quad {\partial w \over \partial t}=G(x,t)$$
Is this correct? Or is there more to it that this.
iii. Using the Frobenius theorem, show that the preceding system has a solution if $\dot a^2(t)=1$ (you don’t need to write down the solution explicitly).
I'm not sure how to approach this part.
