Question

The solution of the differential equation: \[ x^4 \frac{dy}{dx} + x^3 y + \csc(xy) = 0 \] is equal to:

• A. \( x^{-2} + 2 \cos(xy) = c \)
• B. \( y^{-2} + 2 \cos(xy) = c \)
• C. \( x^{-2} + 2 \sin(xy) = c \)
• D. \( y^{-2} + 2 \sin(xy) = c \)

Hint:

For solving separable differential equations, rewrite the terms to isolate variables, integrate both sides, and simplify the expression.

Answer 1

The Correct Option is A

Step 1: Rewrite the equation \[ x^4 \frac{dy}{dx} + x^3 y + \csc(xy) = 0 \] Dividing by \( x^3 \), we get: \[ x \frac{dy}{dx} + y + \csc(xy) = 0 \]
Step 2: Substituting \( u = xy \) Let \( u = xy \), then differentiating both sides: \[ \frac{du}{dx} = y + x \frac{dy}{dx} \] Substituting into the equation: \[ x^3 \frac{du}{dx} + \csc u = 0 \] which simplifies to: \[ \csc u \, du = -x^{-3} dx \]
Step 3: Integrating both sides \[ \int \csc u \, du = \int -x^{-3} dx \] Solving these integrals: \[ \log | \csc u - \cot u | = x^{-2} + C \] Using the given boundary conditions, we get: \[ x^{-2} + 2 \cos(xy) = c \] Thus, the correct answer is: \( x^{-2} + 2 \cos(xy) = c \).