91Ó°ÊÓ

The first important step in solving the given differential equation involves the technique called 'Separation of Variables'. This method allows us to rewrite the equation so that all terms involving one variable are on one side, and the terms involving the other variable are on the opposite side. In this exercise, the goal is to isolate the terms involving \( y \) with \( dy \) and terms involving \( x \) with \( dx \).

• This restructuring turns the equation into a format where each variable can be integrated independently.
• Separate the equation to create two individual integral expressions.

This makes the equation much more manageable, setting the stage for integration of each variable independently, hence simplifying the complex form of the original equation. Once separated, the differential equation from the exercise became \( \mathrm{e}^{(\mathrm{y}/\mathrm{x})} \frac{\mathrm{d}\mathrm{y}}{\mathrm{y}} = \cos \mathrm{x} \cdot \mathrm{d} \mathrm{x} \). This format is essential for subsequent integration.

The process after separating variables is to integrate both sides of the equation. Integration is the method used to solve a differential equation after it has been rearranged correctly. This involves determining the antiderivative of each side.

• For the left side \( \int \mathrm{e}^{(\mathrm{y}/\mathrm{x})} \frac{\mathrm{d}\mathrm{y}}{\mathrm{y}} \), we look for a function whose derivative gives us the expression under the integral sign.
• For the right side \( \int \cos \mathrm{x} \cdot \mathrm{d} \mathrm{x} \), we find an antiderivative of the cosine function.

The integration process simplifies to finding the integral of \( \cos \mathrm{x} \) which is straightforward: its antiderivative is \( \sin \mathrm{x} + C \), where \( C \) is a constant of integration, reflecting the indefinite nature of integration without specific bounds. For the equation in question, integrating both sides gave \( \mathrm{e}^{(y/x)} = \sin x + C \), successfully leading us to the general solution.

After performing the separation of variables and integrating both sides of the differential equation, we reach the 'General Solution'. The general solution comprises all possible solutions of the differential equation and includes an arbitrary constant \( C \), representing the family of curves that fit the equation.

• In this context, \( \mathrm{e}^{(y/x)} = \sin x + C \) is the expression that carves out this relationship.
• The arbitrary constant \( C \) accounts for any specific initial or boundary conditions that might be imposed later.

Determining the general solution is crucial as it encapsulates the broadest scope of outcomes possible for the differential equation. Once a particular solution is identified with more data, \( C \) can take a definite value, but until then, the general solution provides the frame within which all particular solutions will reside. This solution also often needs to be checked against provided options to ensure it matches correctly, as was illustrated in the exercise with the correct choice being \( e^{(y / x)}=\sin x + c \). This check ensures that there are no arithmetic errors in previous steps and that the integrity of the solution is maintained.