Subject: Calculus

Differential Equation

Before we go into the applications of differential equations, let us give a formal definition of differential equation first. Differential equation is a mathematical equation or an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders.

Among the many concepts of calculus, differential equations can be considered as the most important. This has helped various scientist and analyst in modeling phenomena that they are studying. Though it is often impossible to find an explicit formula for differential equations, graphical and numerical approaches can give us the needed information.

Modeling with Differential Equations

In formulating a mathematical model of a real-world problem can either be made through intuitive reasoning about the phenomenon or from a physical law based on evidence from experiments. The mathematical model often takes the form of a differential equation, that is, an equation that contains an unknown function and some of its derivatives. This is not surprising because in a real-world problem we often notice that changes occur and we want to predict future behavior on the basis of how current values change.

Let us examine some examples of how differential equations are used to model physical phenomena.

Models of Population Growth

One model for the growth of a population is based on the assumption that the population grows at a rate proportional to the size of the population. That is a reasonable assumption for a population of bacteria or animals under ideal conditions (unlimited environment, adequate nutrition, absence of predators, and immunity from disease). For the population growth, let us first identify our key variables. We let t represent an independent variable (time) and P which represents the number of individuals in a population. P here becomes a dependent variable.

The rate of growth of population is the derivative of P with respect to time or simply dP/dt. We can therefore write the relation of the population size to the rate of growth of the population as

\frac{dP}{dt}=kP

where k is the proportionality constant. The above equation is our first model of growth; it is a differential equation because it contains an unknown function P and its derivative dP/dt. As we can observe in the above equation, the rate of growth of the population increases as the population increases and vice versa.

Model for the Motion of the Spring

For the next model, let us consider a physics problem. Suppose we have an object of mass m at the end of a vertical spring as shown below.

According to the famous Hooke’s Law, if the string is stretched or compressed by an amount x length from its original length, then it exerts a force that is proportional to x or simply put:

restoring force= -kx

where k is a positive constant (called the spring constant). If we ignore any external resisting forces (due to air resistance or friction) then, by Newton’s Second Law (force equals mass times acceleration), we have)

m\frac{d^2x}{dt^2}=-kx

This is an example of what is called a second-order differential equation because it involves second derivatives. Rewriting the above equation to find its solution, we’ll have

\frac{d^2x}{dt^2}=\frac{-k}{m}x

which says that the second derivative of x is proportional to x but has the opposite sign. The sine and cosine functions have this kind of behaviors and in fact, all solutions of the above equation can be written as combinations of both sine and cosine functions. We could expect this since the spring oscillates which is one property of trigonometric functions.

Illustration

Show that every member of the family of functions

y=\frac{1+ce^t}{1-ce^t}

is a solution of the differential equation

y'=\frac{1}{2}\left (y^2-1\right )

Solution

Using the Quotient Rule to differentiate the expression of y, we’ll have

y'=\frac{\left ( 1-ce^t\right )\left (1+ce^t \right )-\left (1+ce^t \right )\left (-ce^t \right )}{\left ( 1-ce^t\right )^2}
=\frac{2ce^t}{\left (1-ce^t\right )^2}

The right side of the differential equation becomes

y'=\frac{1}{2}\left ( y^2-1 \right )= \frac{1}{2}\left [\left (\frac{1+ce^t}{1-ce^t}\right )^2-1 \right ]
=\frac{1}{2}\left [\frac{\left (1+ce^t\right )^2 - \left (1- ce^t\right )^2}{\left (1-ce^t \right )^2}\right ]
=\frac{2ce^t}{\left (1-ce^t \right )^2}

Therefore, for every value of c, the given function is a solution of the differential equation.

When applying differential equations, we are usually not as interested in finding a family of solutions (the general solution) as we are in finding a solution that satisfies some additional requirement. In many physical problems we need to find the particular solution that satisfies a condition of the form y(t_0 )= y_0. This is called an initial condition, and the problem of finding a solution of the differential equation that satisfies the initial condition is called an initial-value problem.

Geometrically, when we impose an initial condition, we look at the family of solution curves and pick the one that passes through the point (t_0,y_0). Physically, this corresponds to measuring the state of a system at time and using the solution of the initial-value problem to predict the future behavior of the system.

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