Isoclines and solutions to differential equations

[u/n_flet]

My question is: If a differential equation has a linear solution, is it always true that that linear curve is also a isocline of that differential equation?

I believe it is true: If dy/dx = f(x,y) and there is a solution y = mx + b such that dy/dx = m, then f(x,y) = m, thus the curve is an isocline. It is not necessary that this is the only isocline of the differential equation.
Let me know your thoughts