Q.A particle is moving along X-axis. Its acceleration at time $t$ is proportional to its velocity at that time. Find the differential equation of the motion of the particle.

Beyond growth, decay and cooling, a differential equation can capture any situation where a rate of change is described in words and the underlying quantity is the actual object of interest — for example, water rising in a cone at a rate tied to the wetted curved-surface area, a raindrop evaporating at a rate proportional to its own surface area, a balloon's volume inflating at a constant rate, or a person's assets shrinking at a rate proportional to the square root of the current assets. Solving these problems has two distinct stages: first, translating the verbal description into a differential equation by identifying which rate (dV/dt, dh/dt, dr/dt, dA/dt, ...) is being described and what it is stated to be proportional (or equal) to, often requiring an extra geometric relation (such as similar triangles linking a cone's water radius to its height, or t …