The differential equation $y' = \frac{\ln(x^2+y^2)}{x^2 + y^2}$

Question

In my university, the integral calculus teacher gave me this differential equation to solve.

$$ y' = \frac{\ln(x^2+y^2)}{x^2 + y^2} $$

I don't have any clue of what form the solution of this differential equation has.

Answer 1

I would say to simply draw the vector field. Inside the unit circle, $y'$ is negative, so the flow vectors are down and to the right. On the unit circle, $y'=0$, so the flow vectors are horizontal there. Outside the unit circle, the flow vectors are up and to the right (though they become close to horizontal far from the unit circle). So the solutions are essentially constant functions (increasing very slowly) for very positive and very negative initial values of $y$. For intermediate initial values of $y,$ we will see an upward slope approaching the unit circle from left to right, a downward dive passing through the unit circle, and then an upward slope again after leaving the unit circle. There is one solution, symmetric under a $180^\circ$ rotation, where $y\rightarrow 0$ as $x\rightarrow\pm\infty$.

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Update. The attached figure shows approximately what the solutions look like. I misstated the behavior of the symmetric solution; it does not necessarily approach $y=0$ as $x\rightarrow\pm\infty$, but rather has two asymptotes, $y=\pm y_\infty$.