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: Equation Rearrangement \[x^4 \frac{dy}{dx} + x^3 y + \csc(xy) = 0\] Divide by \( x^3 \): \[x \frac{dy}{dx} + y + \csc(xy) = 0\]
Step 2: Substitution \( u = xy \) Let \( u = xy \). Differentiate with respect to \( x \): \[\frac{du}{dx} = y + x \frac{dy}{dx}\] Substitute into the equation: \[x^3 \frac{du}{dx} + \csc u = 0\] Simplify: \[\csc u \, du = -x^{-3} dx\]
Step 3: Integration Integrate both sides: \[\int \csc u \, du = \int -x^{-3} dx\] The integrals yield: \[\log | \csc u - \cot u | = x^{-2} + C\] Applying boundary conditions results in: \[x^{-2} + 2 \cos(xy) = c\] The solution is: \( x^{-2} + 2 \cos(xy) = c \).