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How do I solve the PDE $\frac{\partial^2 \phi}{\partial\eta\partial\xi}=\frac{\partial \phi}{\partial \eta}+\frac{1}{3}$?

What I thought was only to integrate, $\int\frac{\partial^2 \phi}{\partial\eta\partial\xi}d\eta=\int(\frac{\partial\phi}{\partial\eta}+\frac{1}{3})d\eta\Rightarrow\frac{\partial\phi}{\partial\xi}=\phi+\frac{\eta}{3}+f(\xi)$, but how can I proceed from here, if $f$ is an unknown function?

amWhy
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Rebeca Lie Yatsuzuka Silva
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    It would be great if you can update your question with relevant context (source of the problem, your your ideas about solution, your background). Questions which are just problem statements are discouraged here. – Paramanand Singh May 28 '21 at 14:07
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    Your work is correct to this point. $f$ can be any (reasonable) function of $\xi$, so you need some information about initial/boundary conditions to determine $f$ and therefore complete the solution. That said, you could write the solution implicitly. It is a first order differential equation in $\xi$, so you can use integrating factors, leaving $f$ as undetermined. – Cameron Williams May 28 '21 at 14:21

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