Question

Find the general solution to the differential equations. $$ y^{\prime}=e^{-y} \sin x $$

Short answer

The general solution is \( y = \ln(-\cos x + C) \).

Step-by-step solution

Separate Variables

Start by separating the variables for the differential equation \( y' = e^{-y} \sin x \). This can be done by writing the equation as \( \frac{dy}{dx} = e^{-y} \sin x \), and then rearranging to get \( e^{y} \, dy = \sin x \, dx \).

Integrate Both Sides

Integrate both sides of the equation separately. The left side becomes \( \int e^{y} \, dy \) and the right side \( \int \sin x \, dx \).

Solve Left Integral

The integral \( \int e^{y} \, dy \) equals \( e^y + C_1 \), where \( C_1 \) is an integration constant.

Solve Right Integral

The integral \( \int \sin x \, dx \) is equal to \( -\cos x + C_2 \), where \( C_2 \) is another integration constant.

Combine Integrals

Combine the results from both sides to form the equation \( e^y = -\cos x + C \), where \( C = C_1 - C_2 \) is a constant representing the combination of the integration constants.

Solve for y

To express the solution explicitly in terms of \( y \), take the natural logarithm of both sides to obtain \( y = \ln(-\cos x + C) \).

Key concepts

Separation of Variables

The technique of separation of variables is a powerful method used to solve differential equations. It works by rearranging the given equation so that all terms containing the dependent variable and its derivatives appear on one side, while all terms containing the independent variable appear on the other side.
In our example with the differential equation \( y' = e^{-y} \sin x \), we start by expressing the derivative \( y' \) as \( \frac{dy}{dx} \). This transforms the equation into \( \frac{dy}{dx} = e^{-y} \sin x \).
Next, we rearrange to isolate the variables: all \( y \)-related terms should be moved to one side and all \( x \)-related terms to the other. We multiply both sides by \( dx \) and by \( e^{y} \) to get \( e^{y} \ dy = \sin x \ dx \).
This separation allows us to integrate each side with respect to its respective variable. This step lays the foundation for solving the differential equation because it simplifies the problem into two manageable integrals.

Integration of Exponential Functions

When integrating exponential functions, the process can be relatively straightforward, especially with base \( e\). In our problem, we need to integrate \( \int e^{y} \, dy \).
The integral of \( e^{y} \, dy \) is simply \( e^{y} \), given that the derivative of \( e^{y} \) is \( e^{y} \). We must not forget to add an integration constant, resulting in \( e^{y} + C_1 \).
This integration constant \( C_1 \) appears due to the indefinite nature of the integral and plays a critical role when we combine solutions from both sides of a differential equation.
This step is crucial for forming part of the general solution, as it reflects the family of solutions depending on the initial conditions.

Integration of Trigonometric Functions

Integrating trigonometric functions can be intuitive once certain fundamental integrals are familiar. For \( \int \sin x \, dx \), the integral is known to be \( -\cos x \).
This is because the derivative of \( -\cos x \) is \( \sin x \), making \( -\cos x + C_2 \) the antiderivative of \( \sin x \).
As with the exponential integration, an integration constant \( C_2 \) is necessary here. It represents an arbitrary constant of integration arising from the indefinite integral.
Understanding these fundamental integrals and their derivatives is key to solving differential equations and finding their general solutions. These integrations provide pieces of the final solution.

General Solution of Differential Equations

The general solution to a differential equation incorporates all possible particular solutions. By combining the results of our separations and integrations, we form this encompassing solution. In our example, following integration, we have two equations: \( e^y = -\cos x + C_1 \) from the left side, and \( -\cos x + C_2 \) from the right side.
To express the differential equation's complete solution, these are combined to get \( e^y = -\cos x + C \), where \( C \) simplifies as the difference between the initial constants, \( C_1 - C_2 \).
We then solve for \( y \) by taking the natural logarithm of both sides, leading to \( y = \ln(-\cos x + C) \).
This equation \( y = \ln(-\cos x + C) \) defines the general solution, representing any curve that can satisfy the original differential equation given various initial conditions. The integration constant \( C \) encapsulates any specific conditions that define particular solutions.