Problem 22 - Advanced Engineering Mathematics

Question

In Problems 1-22, solve the given differential equation by separation of variables. $$ \left(e^{x}+e^{-x}\right) \frac{d y}{d x}=y^{2} $$

Step-by-Step Solution

Answer

The solution is \( y = -\frac{2}{x + 2C} \).

Step 1: Rewrite the Differential Equation

We start with the differential equation \( \left(e^{x} + e^{-x}\right) \frac{dy}{dx} = y^{2} \). We can rewrite this as \( \frac{dy}{dx} = \frac{y^2}{e^x + e^{-x}} \) to separate the derivative from the algebraic terms.

Step 2: Separate the Variables

To separate the variables, we move all terms involving \( y \) to one side and all terms involving \( x \) to the other. We rewrite the equation as \( \frac{dy}{y^2} = \frac{dx}{e^x + e^{-x}} \).

Step 3: Integrate Both Sides

Integrate each side separately. For the left side, \( \int \frac{dy}{y^2} = -\frac{1}{y} + C_1 \). For the right side, \( \int \frac{dx}{e^x + e^{-x}} = \frac{x}{2} \). So, after integration we have: \( -\frac{1}{y} = \frac{x}{2} + C \).

Step 4: Solve for \( y \)

Rearrange the equation after integration to solve for \( y \). From \( -\frac{1}{y} = \frac{x}{2} + C \), we find \( y = -\frac{1}{\frac{x}{2} + C} \), which simplifies to \( y = -\frac{2}{x + 2C} \) after setting \( 2C = C' \).

Key Concepts

Separation of Variables

Separation of variables is a fundamental technique for solving certain differential equations. The idea is simple: if a differential equation contains two variables, we can "separate" them, grouping all terms involving one variable on one side of the equation and all terms involving the other variable on the opposite side. This is useful because it transforms the equation from a complex mixture of variables into two integrals we can solve individually. For example, in the context of the original problem, we begin by rewriting the equation such that the derivative is isolated:

• Original: \( \left(e^{x} + e^{-x}\right) \frac{dy}{dx} = y^{2} \)
• Rewritten: \( \frac{dy}{dx} = \frac{y^2}{e^x + e^{-x}} \)

This makes it easier to see how we can divide the terms to separate \( y \) and \( x \). After separation, we have:

• \( \frac{dy}{y^2} = \frac{dx}{e^x + e^{-x}} \)

Each side of this equation now deals with only one variable, paving the way for integration.

Integration

Integration is the next step after separating variables in a differential equation. The process involves finding the antiderivative or the integral of each side of the equation after separation. Each side becomes an integral that is often simpler to solve than the original equation. In our example:

• The left side becomes \( \int \frac{dy}{y^2} \).
• The right side becomes \( \int \frac{dx}{e^x + e^{-x}} \).

Integrating the left side \( \int \frac{dy}{y^2} \) gives us \(-\frac{1}{y} + C_1 \), where \( C_1 \) is a constant of integration. For the right side, the integral \( \int \frac{dx}{e^x + e^{-x}} \) can be solved using substitution or recognizing it as a common integral, resulting in \( \frac{x}{2} + C_2 \). We then combine the constants into a single constant \( C \) for simplicity, reaching the integrated form of our equation.

Algebraic Manipulation

Algebraic manipulation comes into play once we have integrated both sides of the differential equation. This involves rearranging terms to solve explicitly for the dependent variable, often \( y \) in these scenarios. After performing the integration in our original equation, we have:

• \( -\frac{1}{y} = \frac{x}{2} + C \)

To solve for \( y \), start by multiplying through by \(-1\) which yields \( \frac{1}{y} = -\frac{x}{2} - C \). Taking the reciprocal of both sides gives \( y = -\frac{1}{\frac{x}{2} + C} \). In many cases, such as this, simplifying further by introducing a new constant can make the solution clearer. Setting \( 2C = C' \) allows us to simplify the equation to:

• \( y = -\frac{2}{x + 2C} \)

This is the final solution, clearly expressing \( y \) in terms of \( x \), demonstrating the power of separating variables and careful algebraic manipulation.