Question

If \( x^4 \, dy + (4x^3 y + 2 \sin x) \, dx = 0 \) and \( f\left( \frac{\pi}{2} \right) = 0 \), then find the value of \( \pi^4 f\left( \frac{\pi}{3} \right) \) (where \( y = f(x) \)):

• A. 81
• B. 80
• C. 83
• D. 9

Hint:

To solve such differential equations, separation of variables and integration are key steps. Make sure to apply any given initial conditions to determine constants.

Answer 1

The Correct Option is A

Step 1: Given Equation.
The given equation is: \[ x^4 \, dy + (4x^3 y + 2 \sin x) \, dx = 0 \] We can rewrite this as: \[ \frac{dy}{dx} = -\frac{4x^3 y + 2 \sin x}{x^4} \] which simplifies to: \[ \frac{dy}{dx} = -\frac{4y}{x} - \frac{2 \sin x}{x^4} \]
Step 2: Separation of Variables.
We separate variables to integrate: \[ \frac{dy}{dx} = -\frac{4y}{x} - \frac{2 \sin x}{x^4} \] Integrating both sides will give us the solution for \( f(x) \).
Step 3: Apply Initial Condition.
Using the condition \( f\left( \frac{\pi}{2} \right) = 0 \), we solve for the constant of integration and find the value of \( f(x) \).
Step 4: Final Answer.
Substituting the value \( \frac{\pi}{3} \) into the function \( f(x) \) and multiplying by \( \pi^4 \), we get: \[ \pi^4 f\left( \frac{\pi}{3} \right) = 81 \] Final Answer: \[ \boxed{81} \]