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If we have this equation:$$\frac{df}{dx} + \frac{dg}{dy}=f+g$$Is this an ODE? If yes, can we consider it linear?

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    $\begingroup$ An ODE can only contain derivatives with respect to one independent variable. $\endgroup$ Oct 17, 2018 at 7:52

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Suppose that $I$ is an intervall in $\mathbb R$, $f,g:I \to \mathbb R$ are differentiable and that $f'(x)+g'(y)=f(x)+g(y)$ for all $x,y \in I$.

Then we have $f'(x)-f(x)=g(y)-g'(y)$ for all $x,y \in I$.

Thus there is a constant $c$ such that

$f'(x)-f(x)=c$ for all $x \in I$ and $g(y)-g'(y)=c$ for all $y \in I$.

Can you proceed ?

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