Question

With Exponential Functions $$d y=e^{-x} d x$$

Short answer

The solution of the differential equation \(dy = e^{-x}dx\) is \(y = -e^{-x} + C\), where C is the constant of integration.

Step-by-step solution

Understanding the Equation

The given equation is a differential equation, which indicates that the change in the variable y, denoted as dy, is equal to the exponential function e raised to the power of negative x, then multiplied by the differential change in x, denoted as dx.

Integrating Both Sides

To solve for y, we need to integrate both sides of the equation with respect to their respective variables. Integrate the left side with respect to y, and the right side with respect to x.

Integration of the Left Side

The integral of dy is simply y, since the integral of 1 with respect to y is y.

Integration of the Right Side

The integral of \(e^{-x}\) with respect to x is \(-e^{-x}\), using the basic rules of integration for exponential functions.

Combining the Integrals

After integrating both sides, we combine the integrals to form the equation \(y = -e^{-x} + C\), where C represents the constant of integration.

Solution of the Differential Equation

The solution of the differential equation is \(y = -e^{-x} + C\), which represents a family of curves, each corresponding to a different value of the constant C.

Key concepts

Exponential Functions

Exponential functions are a fundamental class of functions that feature a constant base raised to a power that is a variable. A typical form of an exponential function is expressed as f(x) = a^x, where a is a positive real number, not equal to one, and x represents the variable. These functions are characterized by the rate of change being proportional to the value of the function itself.

For instance, the function e^x, is an exponential function where the base e is an irrational and transcendental number approximately equal to 2.71828. This particular base is of great importance in mathematics due to its unique properties in calculus, especially concerning the rate of growth. Exponential functions with the base e are used to describe processes that change exponentially, such as radioactive decay and continuous growth scenarios.

In our exercise, e^{-x} represents an exponential function where the variable x is in the exponent, and the negative sign indicates a decay or decrease as x increases. When the variable x changes, the rate of change of the function e^{-x} is directly linked to its current value, making it a dynamic and interesting function to study within differential equations.

Integration

Integration is one of the two main operations in calculus, with the other being differentiation. It can be thought of as the process of calculating the area under a curve described by a function or, more generally, finding the total accumulation of quantities.

When we integrate a function, we are essentially summing up infinitesimally small quantities over an interval to find an accumulated total. Mathematically, this is represented by the integral sign ∫ followed by a function and a differential, for example, ∫f(x)dx. The result of an integration is known as an integral, and it represents a new function that provides the accumulated area under the initial function.

In our case, integrating e^{-x} involves finding a function whose derivative is e^{-x}. This reverses the operation of differentiation. From the steps outlined earlier, we identified through integration rules that the integral of e^{-x} with respect to x is -e^{-x}. This process is crucial for solving differential equations, as it allows us to move from a rate of change (expressed in the differential equation) to the function itself that describes the quantity changing.

Constant of Integration

The constant of integration is an essential aspect of indefinite integrals. Whenever we integrate a function without specific boundaries or limits, the integral is not unique; rather, there is an infinite number of possible functions, all differing by a constant, that would have the same rate of change. This is where the constant of integration, typically denoted as C, comes into play.

It's introduced as a way to represent the family of all possible antiderivatives. In the step-by-step solution for our example, after integrating the right side, we found -e^{-x}, but to account for all possible antiderivatives, we add C to our solution, resulting in y = -e^{-x} + C. This inclusion means that there isn't just one solution to our differential equation but an entire family of solutions, with each member of this family corresponding to a different initial condition represented by C.

Understanding the role of the constant of integration is crucial when dealing with differential equations as it ensures we encompass all potential scenarios and initial conditions in our solution, respecting the principles of calculus that tell us the antiderivative is indeed not a single function but a set of functions.