Problem 43 - Calculus Single Variable

Question

Solve the given differential equation. $$ \left(4+e^{2 x}\right) \frac{d y}{d x}=y e^{x} $$

Step-by-Step Solution

Answer

The solution to the differential equation is \( y = C \sqrt{4 + e^{2x}} \).

Step 1: Separate variables

The given differential equation is \( (4+e^{2x}) \frac{dy}{dx} = y e^x \). First, separate the variables \( y \) and \( x \) by rewriting it as \( \frac{dy}{y} = \frac{e^x}{4+e^{2x}} \, dx \).

Step 2: Integrate both sides

Integrate both sides of the equation separately. This gives \( \int \frac{dy}{y} = \int \frac{e^x}{4+e^{2x}} \, dx \). The left side integrates to \( \ln |y| \).

Step 3: Simplify the right side integral

To integrate the right side, use substitution. Let \( u = 4+e^{2x} \), then \( du = 2e^{2x} \, dx \) or \( e^x \, dx = \frac{du}{2e^x} \). Rewrite \( e^x \) as \( \sqrt{u-4} \), making the integral \( \frac{1}{2} \int \frac{1}{u} \, du \).

Step 4: Complete the integration

The integral \( \frac{1}{2} \int \frac{1}{u} \, du \) yields \( \frac{1}{2} \ln |u| + C \). Substituting back for \( u \), we get \( \frac{1}{2} \ln |4 + e^{2x}| + C \).

Step 5: Solve for \( y \)

Equating both integrals gives \( \ln |y| = \frac{1}{2} \ln |4 + e^{2x}| + C \). Solve for \( y \) by exponentiating both sides to get \( |y| = e^C \sqrt{4 + e^{2x}} \). Thus, \( y = Ce^{1}\sqrt{4 + e^{2x}} \) where \( C = e^C \) is a constant.

Key Concepts

Variable Separation

Variable separation is a common technique used to solve differential equations. It involves rearranging the equation to isolate variables on opposite sides, enabling easier integration. In our problem, this technique allows us to rewrite the equation so the variables are neatly separated:

• The original equation is: \[ \left(4+e^{2x}\right) \frac{d y}{d x} = y e^{x} \]
• By separating variables, we get: \[ \frac{dy}{y} = \frac{e^x}{4+e^{2x}} \, dx \]

This separation helps tremendously when integrating as each side of the equation is prepared individually for integration. Variables that are connected to the two functions are easier to integrate when they are completely isolated from one another. This is the beauty of variable separation: it simplifies complex interactions into manageable pieces.
Remember, the core goal is isolating derivatives and corresponding variables on either side wherever possible, leading to functions that you can handle individually.

Integration Techniques

Integration plays a fundamental role in solving differential equations after variable separation. Here, we must integrate both sides of the separated equation:

• The left side becomes:\[ \int \frac{dy}{y} = \ln |y| + C \]
• For the right side, it involves more complex handling, resulting in:\[ \int \frac{e^x}{4+e^{2x}} \, dx \]

Integration requires understanding different techniques such as direct integration, substitution, and even partial fractions when necessary. Proper application ensures the equation remains balanced and no steps are overlooked. The integration of \( \frac{1}{y} \) yields a logarithmic function, while on the other side, we use substitution for a trickier integration process.
These techniques expand our toolkit, making it easier to solve a wider range of problems, especially in cases where simple integration formulations don’t work. By working through each component carefully, you ensure a correct and elegant solution.

Substitution Method

The substitution method is vital in solving integrals that are complex or involve non-standard functions. In our problem, the integral \( \int \frac{e^x}{4+e^{2x}} \, dx \) is not straightforward and needs substitution to simplify. Here's how substitution is applied:

• First, we choose a substitution: let \( u = 4 + e^{2x} \)
• This means, \( du = 2e^{2x} \, dx \), rearranging gives \( e^x \, dx = \frac{du}{2e^x} \)
• Rewriting the integral, it becomes:\[ \frac{1}{2} \int \frac{1}{u} \, du \]

This change of variables streamlines the process, reducing the integral to a form that is much easier to handle. The integration then becomes simple as using the natural logarithm. After integrating, you reverse substitute the original expression back,leading to the completed integration that fits within the context of the original equation.
The substitution method is a critical strategy, especially in complex integrals where standard techniques would be difficult or impossible to use directly. It transforms the problem into one that is manageable, demonstrating how mathematical insight can simplify complicated expressions.