Drawing a solution curve at a point that is between 2 horizontal asymtote?

Question

I was wondering how would one draw a solution curve that passes through a point sandwich (but spaced apart) between $2$ horizontal asymptotes? This is assuming that the gradient between the $2$ are constant so a linear line. I inserted an image from Khanacademy below to help with the visualisation, but I'm thinking how would I draw the solution curve that, for example passes through $(0,2)$ when every gradient is the same. In the image below, the curve plateau as y approaches $4$ and continues to approach the gradient of $0$ as $x$ increases to infinity. However, if all gradients was equalled to $3$ between $0<y<4$, would the solution curve be a linear line of $y= 3x$ that passes through $(0,2)$ just stop abruptly before hitting $y=0,y=4$? since there's no sign of the gradient decreasing to illustrate a plateau? Feels a bit off when drawing it out, since at $y=0$ and $y=4$, it represents at change from increasing to decreasing (vice versa) right? Thanks

Answer 1

Quite simply since it is a first order differential equation if you are given one point on the curve then the entire curve is completely defined. Solving the equations gives $$\frac{|y|}{|4-y|}=ce^{\frac23x}$$

You put $(0,2)$ to find $c$ and plot $$\frac{|y|}{|4-y|}=e^{\frac23x}$$ which looks like https://www.desmos.com/calculator/dlhx9lk2jn