A differential equation involves an unknown function and its derivative.

Can you think of a function whose derivative is a constant multiple of itself?

We can show that this is the only type of solution of this differential equation:

Therefore, the only solutions of the differential equation are the exponential functions y(t) = y(0)ekt.

Example 1: A bacteria culture starts with 1000 bacteria, and after 2 hours the population is 2500 bacteria. Assuming that the culture grows at a rate proportional to its size, find the population after 6 hours.

dyky

dt

dyky

dt

Differential Equations

Example 2: The half-life of radium-226 is 1590 years. This means that the rate of decay is proportional to the amount present, and half of any given quantity will disintegrate in 1590 years.

(a) Find the formula for the mass of a 100 mg sample of radium-226 that remains after t years.

(b) Find the mass after 100 years.

(c) When will the mass be reduced to 30 mg?

Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings (provided the difference is not too large). Example 3: A hard-boiled egg at 98°C is put in a pan under running 18°C water to cool. After 5 minutes, the egg's temperature is found to be 38°C. How much longer will it take the egg to reach 20°C?