When ODE tells more than the explicit solution.

I would use mathematical models with biological application. Their ODE solutions are often very complex and many of them do not even have closed form solution; however, the formulation of the ODE itself is very intuitive. For instance, the logistic equation: $$\frac{dx}{dt} = r x \left(1 - \frac{x}{K} \right)$$ Or a species growth with harvesting, $$\frac{dx}{dt} = r x \left(1 - \frac{x}{K} \right) - hx$$ An interacting predator-prey system: \begin{align} x' & = bx - k_1x - (dxy)\\ y' & = a(dxy) - k_2y \end{align} I think these are fun and the students can formulate their own and study them using phase-plane analysis and geometric mean.

There are examples where the qualitative approach is very helpful.

Newton's Law of Cooling $$ T'=-k(T-A)$$ is a good example.

The temperature of an object changes proportional to the difference of the object temperature and the room temperature.

The heat equation, $$ \frac {\partial u}{\partial t}= -k \frac {\partial^2 u}{\partial x^2}$$ is another example.

The concavity of heat with respect the state indicates, the rate of change of the temperature with respect the time.

That explains the effect of the temperature of the neighboring points on the object.

Nobody has mentioned the simple harmonic oscillator $$y''=\alpha y.$$

It is easy to see why the solutions are sinusoidal and I consider it the next best example after the classic exponential growth equation.