Question

If \( \lim_{x \to \infty} y(x) = \frac{\pi}{2} \), then the solution of \( x^3 \sin y \frac{dy}{dx} = 2 \) is \( \cos y = \)

• A. \( \frac{3}{x^2} \)
• B. \( \frac{1}{x} \)
• C. \( \frac{1}{x^2} \)
• D. \( \frac{2}{x^3} \)

Hint:

Solve the separable differential equation by integrating both sides with respect to their respective variables. Use the given limit condition to determine the value of the integration constant \( C \). After finding \( C \), substitute it back into the general solution to obtain the particular solution.

Answer 1

The Correct Option is C

The given differential equation is \( x^3 \sin y \frac{dy}{dx} = 2 \).
We can rewrite this as \( \sin y \frac{dy}{dx} = \frac{2}{x^3} \).
Separating the variables, we get \( \sin y dy = \frac{2}{x^3} dx \).
Integrating both sides: $$ \int \sin y dy = \int \frac{2}{x^3} dx $$ $$ -\cos y = 2 \int x^{-3} dx $$ $$ -\cos y = 2 \frac{x^{-2}}{-2} + C $$ $$ -\cos y = -\frac{1}{x^2} + C $$ $$ \cos y = \frac{1}{x^2} - C $$ We are given that \( \lim_{x \to \infty} y(x) = \frac{\pi}{2} \).
Taking the limit as \( x \to \infty \) on both sides of the solution: $$ \lim_{x \to \infty} \cos y = \lim_{x \to \infty} \left( \frac{1}{x^2} - C \right) $$ $$ \cos \left( \lim_{x \to \infty} y(x) \right) = 0 - C $$ $$ \cos \left( \frac{\pi}{2} \right) = -C $$ $$ 0 = -C \implies C = 0 $$ Substituting \( C = 0 \) back into the solution, we get: $$ \cos y = \frac{1}{x^2} - 0 $$ $$ \cos y = \frac{1}{x^2} $$