Let the rate of growth dN/dt of a colony of bacteria be proportional to the square root of the number present at any time. If there are no bacteria present at t = 0, how many are there at a later time? Observe here that the routine separation of variables solution gives an unreasonable answer, and the correct answer, N ≡ 0, is not obtainable from the routine solution. (You have to think, not just follow rules!)

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.