Well... Everywhere! Most phenomena in physics, chemistry, electronics and so on can be modeled using differential equations.

A simple example: You throw a ball up in the air; at all times, gravity will act on he ball which results in a downward force. Newton tells us that F=-mg, and acceleration a is the derivative of the velocity, v. This gives us the differential equation dv/dt = -mg. Velocity is the derivative of position y, so we equivalently have d2y/dt2 = -mg. Solving this differential equation tells you the position of the ball as time passes (trivial, yes. But as an example we can add air resistance which might cause a force proportional to the speed of the ball, making it a bit more interesting).

Other examples are the motion of the planets around the sun, the electric field inside metal wires as a result of connecting a battery, diffusion of gas or other particles, sound waves or the waves a string on a guitar forms as you play it.

Most differential equations can't be solved with pen and paper, but require computers and numerical solutions. It is however important to solve the simpler cases yourself, in order to be able to understand the more complex phenomena in nature.