Question

The solution of \( x \sec \left( \frac{x}{y} \right) - y \, dx + x \, dy = 0 \) is:

• A. \( \log |k| - \cos \left( \frac{x}{y} \right) = c \)
• B. \( \log |k| - \cos \left( \frac{x}{y} \right) = c \)
• C. \( \log |k| - \sin \left( \frac{x}{y} \right) = c \)
• D. \( \log |k| - \sin \left( \frac{x}{y} \right) = c \)

Hint:

When solving differential equations, look for substitutions that simplify the expression, such as using trigonometric identities for complex terms.

Answer 1

The Correct Option is B

Step 1: Solve the given differential equation. We solve the equation by using the method of integration and applying the necessary transformations. After solving, we get the solution in the form of a logarithmic expression involving \( \cos \left( \frac{x}{y} \right) \).
Step 2: Conclusion. Thus, the solution to the differential equation is \( \log |k| - \cos \left( \frac{x}{y} \right) = c \).