Nineteenth-century physician Jean Louis Marie Poiseuille...

Nineteenth-century physician Jean Louis Marie Poiseuille discovered that the rate of change of the velocity of blood $r$ cm from the central axis of an artery (in centimeters per second per centimeter) is given by $$a(r)=-k r$$ where $k$ is a constant. If the radius of an artery is $R \mathrm{cm}$, find an expression for the velocity of blood as a function of $r$ (see the accompanying figure).

Nineteenth-century physician Jean Louis Marie Poiseuille discovered that the rate of change of the velocity of blood $r$ cm from the central axis of an artery (in centimeters per second per centimeter) is given by
$$a(r)=-k r$$ where $k$ is a constant. If the radius of an artery is $R \mathrm{cm}$, find an expression for the velocity of blood as a function of $r$ (see the accompanying figure).

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Key Concepts

First-Order Differential Equations

A first-order differential equation relates a function to its first derivative with respect to a variable. In many physical contexts, such as fluid flow, the rate of change is given by a simple function of the spatial variable, and solving the equation involves integrating this derivative to determine the original function.

Integration as the Inverse Process of Differentiation

Integration is used to recover a function from its rate of change. When a derivative is given in terms of the independent variable, integrating that expression yields the original function up to an arbitrary constant. This principle is fundamental in calculus and is widely used to model natural phenomena.

Applying Boundary Conditions

Once an indefinite integral is obtained, the arbitrary constant that arises must be determined by using a boundary condition. By substituting a known value of the function at a specific point, typically at a boundary of the domain, you obtain a unique solution that accurately describes the physical situation.

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