Formation of Partial Differential Equation | Elimination of Arbitrary Functions

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Partial Differential Equation
Formation of PDE by Eliminating Arbitrary Function

Questions
𝑧=1/𝑢(𝑓(𝑢−𝑎𝑡)+𝑓(𝑢+𝑎𝑡))
𝑧=𝑦𝑓(𝑥)+𝑥𝑔(𝑦)
𝑧=𝑦^2 𝑓(𝑥)+𝑥^2 𝑔(𝑦)
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