Applications of Differential Equations Differential Equation are widely applied in the propagation of sound, electric and magnetic field, radio waves, optics, elasticity, spectral analysis of radiation and other scientific field. It is also widely used in the study of demography and applied in economics and finance.
Example Chemical Reaction In a particular chemical reaction, the differential equation that relates the relationship between the quantity y of the substance and the time t is given by Where w is a positive constant. If y=0 when t=0, find the solution for y as a function of x .
Newton Law of Cooling 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. A cake is taken from an oven when its temperature has reached 150 0 c.
i. If y(t) is the temperature of the cake after t minutes, show that by Newton’s Law of Cooling, ii. Find a formula for the temperature of the cake after t minutes iii. If the temperature of the cake is 100 0 C after 10 minutes, what is its temperature after 20 minutes? iv. When will the cake have cooled to 40 0 C
Solution (I) Difference in temperature = y-25 Thus ,where k is a constant (ii) When,
(iii) t = 0 minutes, y = 150 o C When t = 20 minutes
It takes 41.5 minutes for the cake to cool to 40 o C
Exercise Carbon-14 dating is used to determine the age of fossils. It is based on the fact that the amount of carbon-14 in living organism is essentially constant (say 100 units) and that when an organism dies, the rate of change of carbon-14 in the organism is proportional to the amount present. If x is the amount of carbon-14 present in t year, then Where m=constant Subject to the fact that when t=0, x=100, the half-life of carbon-14 is 5600 years. An anthropologists discover a fossil that contains 2 units of carbon-14. Find the age of the fossil.