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Does it just mean that the solution has continuous derivatives up to some desired order? In the context of PDE's, would it just mean that the function is continuously differentiable in some variable up to the highest partial derivative in $x$ for example?

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In my university courses, "smooth" meant something different for each professor/course: either $\mathcal{C}^{\infty}$, or differentiable, or differentiable up to a certain order of derivative. So I think you should check what is, in your context, the definition of "smooth", and use it for the solution of the PDE too.

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Usually smooth refers to a class of functions who's derivative is always continuous, usually denoted $f\in C^\infty$. A common example would be $f(x)=x$ then, $f'(x)=1$ and $\frac{d^n}{dx^n}f(x)=0$ for any $n \geq 2\in \mathbb{N}$. So $f$ is a smooth solution to a PDE if 1. $f$ is a solution and 2. $f$ satisfies the above property.

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