Qualitative study of differential equation

Question

Good morning everyone ,

I'd like to ask the following doubt within the exercise :

Study $y'= \frac{x \cdot e^{y(x)}}{(y(x)-1)} $ .

After separable variables I obtain : $-ye^{-y}=\frac {x^{2}}{2} + C$

The doubt was (in the picture) : I'd like to know after the first two graph on the left (one of $-ye^{-y}$, one of $\frac {x^{2}}{2} + C$) how to properly put them in a sort of bijection, or at least gather them properly so that I could find the graph on the right, that is my final goal.

Help would be appreaciated,

Thanks.

Answer 1

With the help of this MATHEMATICA script gr = Table[ If[c > 0, ContourPlot[y E^-y + x^2/2 + c == 0, {x, -5, 5}, {y, -5, 5}, ContourStyle -> Red], ContourPlot[y E^-y + x^2/2 + c == 0, {x, -5, 5}, {y, -5, 5}, ContourStyle -> Blue]], {c, -1, 1, 0.1}]; Show[gr]

we can verify the results obtained analytically confirming that

$$ ye^{-y}+\frac{x^2}{2}+c = 0 $$

is an implicit function $y=y(x,c)$ only for $c \ge 0$ (blue curves). In red the results for $c < 0$

I hope this helps.