Solving the differential equations that arise from modeling...

Solving the differential equations that arise from modeling may require using integration by parts. [See formula (1).] Morphine is a drug that is widely used for pain management. However, morphine can cause fatal respiratory arrest. since pain perception and drug tolerance vary with patients, morphine is gradually administered with small increments until the pain is controlled or side effects begin to appear. In one intravenous infusion, morphine was injected continuously at an increasing rate of $t$ milligrams per hour. Suppose that the body removes the drug at a rate proportional to the amount of the drug present in the body, with constant of proportionality $k=35 .$ Let $f(t)$ denote the amount of morphine in the body, $t$ hours from the beginning of the infusion. (a) Find a differential equation satisfied by $f(t).$ (b) Assuming that the infusion lasted 8 hours, determine the amount of morphine in the body during the infusion if at the beginning of the infusion the body was free of morphine.

Solving the differential equations that arise from modeling may require using integration by parts. [See formula (1).]

Morphine is a drug that is widely used for pain management. However, morphine can cause fatal respiratory arrest. since pain perception and drug tolerance vary with patients, morphine is gradually administered with small increments until the pain is controlled or side effects begin to appear.

In one intravenous infusion, morphine was injected continuously at an increasing rate of $t$ milligrams per hour. Suppose that the body removes the drug at a rate proportional to the amount of the drug present in the body, with constant of proportionality $k=35 .$ Let $f(t)$ denote the amount of morphine in the body, $t$ hours from the beginning of the infusion.
(a) Find a differential equation satisfied by $f(t).$
(b) Assuming that the infusion lasted 8 hours, determine the amount of morphine in the body during the infusion if at the beginning of the infusion the body was free of morphine.

Key Concepts

Differential Equation Modeling

This concept involves using differential equations to represent real-world phenomena mathematically. In many scenarios, such as in medicine or physics, the rate of change of a quantity is related to the quantity itself or to other influencing factors, forming the basis of a model that can be analyzed and solved to understand the behavior over time.

First Order Linear Differential Equations

These are differential equations of the form dy/dt + P(t)y = Q(t) where the solution can often be found via an integrating factor. They are commonly used in modeling processes with rates that are proportional to the current state and are essential in describing systems with inputs and decay or growth mechanisms.

Exponential Decay

This concept describes a process where the rate of decrease of a quantity is proportional to its current value, resulting in a decay modeled by an exponential function. It is a common feature in models dealing with removal, degradation, or discharge processes.

Integration by Parts

Integration by parts is a technique used to integrate products of functions, based on the product rule of differentiation. It is often employed in solving integrals that arise when dealing with input functions that multiply exponential functions or other factors in the process of finding a particular solution to a differential equation.

Initial Value Problems

An initial value problem specifies the state of the system at a particular time, providing the necessary starting condition to uniquely determine the solution of a differential equation. This is critical in many modeling applications where the state at time zero influences the subsequent behavior of the system.

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