How can I intuitively predict the shape of a solution to ordinary differential equations? I am new to differential equations, and I realized that I don't have even an intuition as to what solutions to ordinary differential equations would roughly look like. For example, given the governing equation:
$$ \frac{\partial^2 y}{\partial x^2} -\frac{\partial y}{\partial x} + y = 0 $$
for the range $$0<x<10$$
given the initial conditions $$y(0) = 1$$
$$y(10) = 5$$
The plotted graph for the function y looks like:

How can you intuitively predict that y will be mostly flat in the range 0<x<6?
If so, how would you intuitively picture the solution for this function y without solving for it analytically?

Slope Fields offer a good visual aid, sometimes almost showing the actual solutions. You only calculate the slope at select points using the given formula for the first derivative, so it's usually a somewhat less intensive process than solving numerically.

Phase Portraits Have similar features and are more generalizable to second order differential equations.

