What is a drawback of trying to solve a partial differential equation explicitly?

As far as I'm aware a PDE can't be solved explicitly because of the nature of the equation. However, you can separate the variables of a PDE essentially turning it into multiple ODEs which then can be solved explicitly The draw back of this is that most terms in a PDE are a combination of variables, such as 3xy+2yz+xyz, rather than being separated neatly already (making separation into ODEs easy), such as x+y+z. In cases where the variables are jumbled up with each other, there are often faster and easier ways of solving the PDE than trying to separate the variables into ODEs and solving explicitly.

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