Question

Solve each differential equation. $$ y^{\prime}+y \tan x=\sec x $$

Short answer

The solution is \( y = \tan{x} + C \cos{x} \), where \( C \) is a constant.

Step-by-step solution

Identify the Differential Equation Type

The given differential equation is \( y' + y \tan{x} = \sec{x} \). It resembles a first-order linear differential equation of the form \( y' + P(x)y = Q(x) \). Here, \( P(x) = \tan{x} \) and \( Q(x) = \sec{x} \).

Compute the Integrating Factor

Calculate the integrating factor \( \mu(x) \) using the formula: \( \mu(x) = e^{\int P(x) \, dx} \). Here, \( \mu(x) = e^{\int \tan{x} \, dx} \). Recall that \( \int \tan{x} \, dx = \ln|\sec{x}| \). Therefore, the integrating factor is \( \mu(x) = e^{\ln|\sec{x}|} = \sec{x} \).

Multiply the Equation by the Integrating Factor

Multiply the entire differential equation by the integrating factor \( \sec{x} \): \( \sec{x}y' + y\sec{x}\tan{x} = \sec^2{x} \).

Simplify to a Perfect Derivative

Notice that the left side of the equation is now a derivative of a product: \( \frac{d}{dx}(y \sec{x}) = \sec^2{x} \).

Integrate Both Sides

Integrate the equation from Step 4: \( \int \frac{d}{dx}(y \sec{x}) \, dx = \int \sec^2{x} \, dx \). This results in \( y \sec{x} = \tan{x} + C \) where \( C \) is the constant of integration.

Solve for y

Solve for \( y \) by dividing both sides by \( \sec{x} \): \( y = \tan{x} + C \cos{x} \). This is the general solution of the differential equation.

Key concepts

First-Order Linear Differential Equations

A first-order linear differential equation is an equation that involves the derivatives of a function and can be expressed in the standard form: \( y' + P(x)y = Q(x) \). In simpler terms, it's a differential equation where the highest derivative is first order, and the coefficients of \( y' \) and \( y \) are functions of \( x \).

In our example, the equation \( y' + y \tan{x} = \sec{x} \) fits this format perfectly. Here, \( P(x) = \tan{x} \) and \( Q(x) = \sec{x} \). Recognizing this form is crucial because it governs the methods we use to solve the equation, specifically, the integrating factor method.

Integrating Factor

The integrating factor is a clever tool used to solve first-order linear differential equations. It is particularly useful because it allows us to transform the equation into a form that is easy to integrate, specifically into a perfect derivative.

To find the integrating factor, we calculate \( \mu(x) = e^{\int P(x) \, dx} \). For our example, where \( P(x) = \tan{x} \), the integrating factor becomes \( e^{\int \tan{x} \, dx} \). Knowing that \( \int \tan{x} \, dx = \ln|\sec{x}| \), we find that the integrating factor is \( \sec{x} \). This multiplying by \( \sec{x} \) adjusts the differential equation into a form that makes integration straightforward.

General Solution

The general solution of a differential equation is a formula expression that encompasses all possible solutions to the equation. In the process of solving, after applying the integrating factor and using integration, we find the solution in terms of \( y \) and include an arbitrary constant, often denoted as \( C \).

For our equation, once adjusted to \( \frac{d}{dx}(y \sec{x}) = \sec^2{x} \), integrating both sides leads to \( y \sec{x} = \tan{x} + C \). Finally, solving for \( y \) results in the general solution: \( y = \tan{x} + C \cos{x} \). This expression not only satisfies the differential equation, but by varying \( C \), it can represent a family of curves.

Constant of Integration

The constant of integration, often symbolized by \( C \), is an integral part of any indefinite integration. In differential equations, it represents the infinite family of solutions derived from an antiderivative.

When we integrate a function like \( \sec^2{x} \) in our differential equation, the constant of integration \( C \) arises from the indefinite nature of integration. Each different value of \( C \) translates to a different particular solution to the differential equation, showing how they fit within the same general formula: \( y = \tan{x} + C \cos{x} \). This flexibility is crucial for initial value problems where specific conditions dictate the exact solution.