Question

Find the general solution to the differential equation.\(y^{\prime}=e^{x} e^{y}\)

Short answer

The general solution is \( y = -\ln(-e^x - C) \).

Step-by-step solution

Write the Differential Equation

The given differential equation is \( y' = e^x e^y \). The goal is to find the general solution for this equation.

Separate Variables

To solve this differential equation, we first need to separate the variables. Divide both sides by \( e^y \) and multiply both sides by \( dx \) to get: \[ \frac{dy}{e^y} = e^x dx \].

Integrate Both Sides

Now, integrate both sides of the equation. The left side is integrated with respect to \( y \), and the right side is integrated with respect to \( x \). This gives us:\[ \int \frac{dy}{e^y} = \int e^x \, dx \]. The integration on the left is \( \int e^{-y} \, dy = -e^{-y} \) and on the right is \( \int e^x \, dx = e^x \).

Solve the Integrals

From the integrations, we obtain:\[ -e^{-y} = e^x + C \] where \( C \) is the constant of integration. Remember, we represent it generally here since the specific initial conditions aren't given.

Solve for y

Rearrange the equation to express \( y \) explicitly. Multiply both sides by \( -1 \) to get:\[ e^{-y} = -e^x - C \] .Take the natural logarithm of both sides to solve for \( y \):\[ -y = \ln(-e^x - C) \]. Finally, multiply by \( -1 \) to solve for \( y \): \[ y = -\ln(-e^x - C) \].

Write the General Solution

The general solution for the differential equation is:\[ y = -\ln(-e^x - C) \]. This represents the general solution of the given differential equation, where \( C \) is an arbitrary constant.

Key concepts

Separation of Variables

Separation of variables is a fundamental step in solving certain types of differential equations. In this technique, you rearrange and isolate the variables so each side of the equation contains only one type of variable. This allows you to integrate with respect to each variable independently.
In the given differential equation, \( y' = e^x e^y \), the goal is to keep all terms involving \( y \) on one side and all terms involving \( x \) on the other. To do this, divide both sides by \( e^y \) and multiply both sides by \( dx \). This manipulation gives us the separated form:

• \( \frac{dy}{e^y} = e^x dx \)

This separation is crucial because it sets up the equation for integration. Remember, this technique works best when the equation can be neatly divided in such a way that each variable is isolated on opposite sides of the equality.

Integration

Integration is a powerful mathematical tool used in calculus to find functions from their derivatives. Once you have successfully separated the variables, the next step is to integrate both sides of the equation. Each side is integrated with respect to its own variable.For the separated equation \( \frac{dy}{e^y} = e^x dx \), you perform the following integrations:

• Left side: \( \int e^{-y} \, dy = -e^{-y} \)
• Right side: \( \int e^x \, dx = e^x \)

Integration provides us with two expressions that are equal to each other, albeit to a constant \( C \), because indefinite integrals include an arbitrary constant of integration.
Here, the equation becomes \( -e^{-y} = e^x + C \). This step bridges the gap between the given differential form and a more concrete algebraic expression involving \( y \).

General Solution

After integrating both sides, you arrive at an expression that still includes the arbitrary constant \( C \). At this stage, you need to solve for \( y \) to express the general solution of the differential equation.Starting from the equation \( -e^{-y} = e^x + C \), the process to isolate \( y \) involves:

• Multiplying both sides by \( -1 \) to clean up the left side: \( e^{-y} = -e^x - C \)
• Taking the natural logarithm of both sides: \( -y = \ln(-e^x - C) \)
• Finally, solving for \( y \) by multiplying through by \( -1 \): \( y = -\ln(-e^x - C) \)

This final expression, \( y = -\ln(-e^x - C) \), is the general solution to the differential equation. It encompasses all possible solutions by incorporating the constant \( C \). The general solution is key to understanding how differential equations describe a family of curves rather than a single solution. It also provides a basis for further exploration, especially when specific initial conditions are given.