Category: Applications of First-Order ODE
Growth and Decay Problems
Let denote the amount of a substance (or population) that is either growing or decaying. If we assume that , the time rate of change of this amount of substance, is proportional to the amount of substance present, we have that , where is the constant of proportionality.
Temperature Problems
Newton's law of cooling, which is equally applicable to heating, states that the time rate of change of the temperature of a body is proportional to the temperature difference between the body and its surrounding medium. Let denote the temperature of the body and denote the temperature of the surrounding medium. Then, the time rate of change of the temperature of the body is , and Newton's law of cooling can be formulated as , or as , where is a positive constant of proportionality. Once is chosen positive, the minus sign is required in Newton's law to make negative in a cooling process, when is greater than , and to make it positive in a heating process, when is smaller than .
Falling Body Problems
Consider a vertically falling body of mass that is being influenced only by gravity and an air resistance that is proportional to the velocity of the body. Assume that both gravity and mass remain constant and, for convenience, choose the downward direction as the positive direction.
Dilution Problems
Consider a tank which initially holds gal. of brine that contains lb. of salt. Another brine solution containing lb. of salt per gallon is poured into the tank at a rate of gal./min., while, simultaneously, the well-stirred solution leaves the tank at a rate of gal./min. The problem is to find the amount of salt in the tank at any time . Let denote the amount (in pounds) of salt in the tank at any time . The time rate of change of , , equals the rate at which the salt enters the tank minus the rate at which the salt leaves the tank. The salt enters the tank at a rate of lb./min. To determine the rate at which the salt leaves the tank, we first calculate the volume of brine in the tank at any time , which is the initial volume plus the volume of brine added minus the volume of brine removed . Thus, the volume of brine at any time is . The concentration of salt in the tank at any time is , from which it follows that the salt leaves the tank at a rate of gal./min.
Electrical Circuits
The basic equation governing the amount of current (in amperes) in a simple RL circuit consisting of a resistance (in ohms), an inductance (in henries), and an electromotive force (abbreviated as 'emf') (in volts) is .For an RC circuit consisting of a resistance, a capacitance (in farads), an emf, and no inductance, the equation governing the amount of electrical charge (in coulombs) on the capacitor is . The relationship between and is .
Orthogonal Trajectories
Consider a one-parameter family of curves in the xy-plane defined by , where denotes the parameter. The problem is to find another one-parameter family of curves called the orthogonal trajectories of the family defined above and given analytically by , such that every curve in this new family intersects at right angles every curve in the original family.