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In finding a solution to $$\begin{cases} u_{tt}=\nabla^2u, & x\in\mathbb R^2, \,t\gt0 \\ u(x,0)=x_1^2, u_t(x,0)=x_1^2+x_2^2, & x\in\mathbb R^2 \end{cases} $$

a hint was given to split it into two 1-D problems and apply d'Alembert's formula. I don't even understand the concept of splitting it. I also don't know any other way to solve this. I do know how to use d'Alembert's formula, so maybe if someone could help me with the 'splitting' I could solve it.

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The idea of splitting refers to the fact that the wave equation is linear: a sum of two solutions is also a solution to the same equation. This makes it possible to assemble a solution that satisfies the initial conditions from two or more pieces. In this case, inspection of the initial conditions shows that both of them are the sum of a function of $x_1$ and a function of $x_2$: $$\begin{align*} u(x,0)& = \color{Red}{x_1^2}\; + \; \color{Blue}{0} \\ u_t(x,0)& = \color{Red}{x_1^2} \;+\; \color{Blue}{x_2^2} \end{align*}$$ So, if you can find a solution for the initial conditions in red, and then a solution for the initial conditions in blue, their sum will give what you want.

The red and blue sub-problems are essentially 1-D because if the initial conditions for the wave equation do not involve some of the $x_j$-variables, the solution will not involve them either. This can be justified by appealing to the translation invariance of the equation and the uniqueness of solutions. But chances are that you are not expected to justify this part.

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