Question

Find the general solution of each of the following differential equations. \(d x+\left(x-e^{y}\right) d y=0\)

Short answer

The general solution is \( x = C e^{-y} + e^y \).

Step-by-step solution

Rewrite the Differential Equation

The given differential equation is: \[ d x + (x - e^y) d y = 0 \]Rewrite this equation as: \[ d x = - (x - e^y) d y \]

Separate Variables

Separate the variables x and y by dividing both sides by \(x - e^y\): \[ \frac{d x}{x - e^y} = - d y \]

Integrate Both Sides

Integrate both sides of the equation. The left side integrates with respect to x, and the right side integrates with respect to y: \[ \int \frac{1}{x - e^y} d x = - \int d y \]Let’s evaluate these integrals.

Solve the Integrals

Consider the integral on the left side. First, substitute \(u = x - e^y\) so \(du = dx\).Then the integral becomes: \[ \int \frac{1}{u} d u = \ln |u| + C_1 = \ln |x - e^y| + C_1 \]The integral on the right side is: \[ - \int d y = - y + C_2 \]

Combine the Results

Combine the integrated results from both sides: \[ \ln |x - e^y| = - y + C \]Where \(C = C_2 - C_1\) is a constant.

Exponentiate and Solve for General Solution

Exponentiate both sides to remove the natural logarithm: \[ |x - e^y| = e^{-y + C} \]This can be written as: \[ x - e^y = \text{constant} \times e^{-y} \]The general solution of the differential equation is: \[ x = C e^{-y} + e^y \]

Key concepts

separation of variables

Separation of variables is a method used to solve differential equations. The goal is to isolate the variables involved so that each variable appears on one side of the equation only. This method works best when you can rewrite the equation in the form where one side is a function of one variable and the other side is a function of another variable.

Let's look at our example equation: \(d x + (x - e^y) d y = 0\).
First, we rewrite it to isolate the term involving dx:

\[ d x = - (x - e^y) d y \]
Next, we divide both sides of the equation to separate the variables:

\[ \frac{d x}{x - e^y} = - d y \].
Here, all the terms with x are on one side (left) and all the terms with y are on the other side (right). This step of separating the variables simplifies the process of integration, leading us to the next step in solving the differential equation.

integration

Integration is the process of finding the integral of a function, and it is an essential step when solving differential equations using the separation of variables method. Once the variables have been separated, the next step involves integrating both sides of the equation.

From our separated variables equation: \[ \frac{d x}{x - e^y} = - d y \], we integrate both sides:
\[ \int \frac{1}{x - e^y} d x = - \int d y \].

For the left side, we use the substitution method. Set \(u = x - e^y\), hence \(du = dx\), the integral becomes:
\[ \int \frac{1}{u} d u = \ln |u| + C_1 \].
Replacing \(u\) back with \(x - e^y\), we get:
\[ \ln |x - e^y| + C_1 \].
For the right side, the integral is straightforward:
\[ - \int d y = - y + C_2 \].
Integrating helps in moving from a differential equation to a more familiar algebraic equation involving constants of integration, which leads us towards the general solution.

general solution

The general solution of a differential equation is a family of solutions that includes an arbitrary constant (or constants). It represents all possible solutions to the differential equation.

Once we have integrated both sides of our example equation, we combine the results:
\[ \ln |x - e^y| = - y + C \], where \(C\) is an arbitrary constant derived from \(C_2 - C_1\).
To solve for \(x\), we exponentiate both sides to get rid of the natural logarithm:
\[ |x - e^y| = e^{-y + C} \].
This can be simplified further by expressing the exponent:
\[ x - e^y = A e^{-y} \], where \(A = e^C\) is another constant.
Finally, the general solution to our differential equation is:
\[ x = A e^{-y} + e^y \]. This reveals an infinite set of solutions depending on the value of the constant \(A\).
Understanding how to derive the general solution is crucial for solving differential equations, as it provides the foundation for finding specific solutions given initial conditions.