Question

Finding a General Solution In Exercises \(41-52,\) use integration to find a general solution of the differential equation. $$ \frac{d y}{d x}=\frac{e^{x}}{4+e^{x}} $$

Short answer

The general solution of the given differential equation is \(4y = x - 4 \ln |4+e^{x}| + C\).

Step-by-step solution

Separate variables

Firstly, aim to separate the variables so that \(y\) and \(x\) are on different sides of the equation. Transform the equation as: \(4 \cdot dy =(e^{x} \cdot dx) / (4+e^{x})\)

Additional transformations

Next, perform some additional transformations, writing \(e^{x}\) as \((4+e^{x}) - 4\), which makes the right side of equation become \(4 \cdot dy = ((4+e^{x})-4 \cdot dx)/ (4+e^{x})\), or \(4 \cdot dy = dx - 4 \cdot dx / (4+e^{x})\)

Integrate both sides

Now integrate both sides of the equation. This results in \(4y = x - 4\int dx/(4+e^{x})\). The integral on the right side could be solved with the substitution method, by setting \(u =4+e^{x}\), so \(du = e^{x}dx\)

Complete the solution

To find the general solution, calculate the integral and return to the variable \(x\). The complete solution is \(4y = x - 4 \ln |4+e^{x}| + C\), where \(C\) is the constant of integration.

Key concepts

Integration

Integration is a fundamental concept in calculus often used to find solutions to differential equations. The idea is to find the antiderivative of a function, which generally means reversing the process of differentiation. Understanding integration helps us determine the solution to a differential equation by finding a function whose derivative is the given function.

For this particular exercise, integration is applied after transforming the original differential equation into a form where the variables can be separated. The result of the integration will include an arbitrary constant, often denoted as \( C \), because integration represents an indefinite process that yields multiple potential solutions.

• Integral calculus reverses differentiation.
• General solutions include a constant \( C \).

Separation of Variables

Separation of Variables is a technique used in solving differential equations, where we separate the variables \( y \) and \( x \), pushing them on opposite sides of the equation. This allows each side of the equation to be linked solely with one variable, providing a pathway to solve the equation using integration.

In the given exercise, initially, the equation \( \frac{d y}{d x} = \frac{e^{x}}{4+e^{x}} \) is manipulated so that all terms involving \( y \) are on one side, and all terms involving \( x \) are on the other.

• Separate \( dy \) and \( dx \) onto different sides.
• Prepare the equation for integration.

Substitution Method

The substitution method is a powerful tool in integrating complex functions. This method involves substituting a part of the integral with a new variable \( u \), simplifying the equation and making it more accessible.

In our exercise, after transforming the equation into a separable form, substitution is used to integrate the expression \( \int \frac{dx}{4+e^{x}} \). To do this, we set \( u = 4 + e^{x} \) and find \( du = e^{x} dx \). This substitution simplifies the integral into a more straightforward form, \( \int \frac{1}{u} du \), which then can be easily integrated to \( \ln |u| \).

• Replace an expression with a single different variable \( u \).
• Simplify the integral for easier computation.
• Don't forget to switch back to the original variable after integration.