Question

If \(y + \frac{d}{dx}(xy) = x(\sin x + \log x)\) then

• A. \(y = \cos x + \frac{2 \sin x}{x} + \frac{2}{x^2} \cos x + \frac{x}{3} \log x - \frac{x}{9} + \frac{c}{x^2}\)
• B. \(y = -\cos x - \frac{2}{x} \sin x + \frac{2}{x^2} \cos x + \frac{x}{3} \log x - \frac{x}{9} + \frac{c}{x^2}\)
• C. \(y = -\cos x + \frac{2}{x} \sin x + \frac{2}{x^2} \cos x + \frac{x}{3} \log x - \frac{x}{9} + \frac{c}{x^2}\)
• D. \(y = \cos x - \frac{2}{x} \sin x + \frac{2}{x^3} \cos x + \frac{x}{3} \log x - \frac{x}{9} + \frac{c}{x^2}\)

Hint:

If the equation contains \(\frac{d}{dx}(xy)\), expand it first. Very often it becomes a standard linear differential equation.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Expand the derivative of the product and convert into a linear differential equation form.
Step 2: Key Formula or Approach:
Equation: \(y + x y' + y = x(\sin x + \log x) \implies x y' + 2y = x(\sin x + \log x)\).
Linear form: \(y' + \frac{2}{x}y = \sin x + \log x\).
Step 3: Detailed Explanation:
IF \(= e^{\int \frac{2}{x} dx} = x^2\).
\(y \cdot x^2 = \int x^2(\sin x + \log x) dx\)
\(= \int x^2 \sin x dx + \int x^2 \log x dx\)
By parts: \(= (-x^2 \cos x + 2x \sin x + 2\cos x) + (\frac{x^3}{3} \log x - \frac{x^3}{9}) + C\).
Divide by \(x^2\) to find \(y\).
Step 4: Final Answer:
Matches option (C).