Application of Differential Equations

LAWS OF GROWTH AND DECAY

This kind of problems start with an IVP (Initial Value Problem) shown below,

where k refers to the constant of proportionality, is used in modeling first-order DEs involving growth and decay. I have learned in the video discussion that the rate of growth in certain populations over a short period of time is directly proportional to the population present during time t. Given some arbitrary initial time to, we can determine the population for some time in the future by solving for constant of proportionality and using a subsequent measurement of x at time t1 > to . Positive k's are used when problems characterize the growth and negative k values or decay problems.

NEWTON'S LAW OF COOLING

Approximating the temperature of an object can be obtained through the use of Newton's Law of Cooling. The temperature of the body changes at a rate that is directly proportional to the difference between the outside medium and the body itself. Thus, it is always assumed that the constant of proportionality will be the same whether the temperature is increasing or decreasing. The mathematical formulation of Newton's law of cooling is given by this differential equation,

where k is the constant of proportionality, T is the temperature of the object and Tm is the temperature of the medium (ambient temperature).

mixture problems

Mixture of two fluids or substances give rise to linear first differential equation. The problem usually starts with a substance dissolved in a liquid, where liquid will enter and leave the tank. The liquid may or may not contain the substance dissolved in it. But the liquid leaving the tank will always contain the dissolve substance. It is always assumed that the concentration of the substance in the liquid is uniform throughout the tank. This differential equation is used when modeling the mixture problems :

SAMPLE PROBLEMS

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