Question

The general solution of \[ \left( x \frac{dy}{dx} - y \right) \sin \frac{y}{x} = x^3 e^x \text{ is:} \]

• A. \( e^x(x-1) + \cos \frac{y}{x} + c = 0 \)
• B. \( x e^x + \cos \frac{y}{x} + c = 0 \)
• C. \( e^x(x+1) + \cos \frac{y}{x} + c = 0 \)
• D. \( e^x x - \cos \frac{y}{x} + c = 0 \)

Hint:

When solving first-order linear differential equations, look for substitutions or methods like integrating factors to simplify the equation.

Answer 1

The Correct Option is A

Step 1: Begin with the given differential equation: \[ \left( x \frac{dy}{dx} - y \right) \sin \frac{y}{x} = x^3 e^x \] Rearrange to: \[ x \frac{dy}{dx} - y = \frac{x^3 e^x}{\sin \frac{y}{x}} \] 

Step 2: Use an appropriate method to solve this equation. Since it is a linear form, we can apply the integrating factor method. After simplifications, integrate both sides with respect to \(x\). 

Step 3: The general solution obtained is: \[ e^x (x-1) + \cos \frac{y}{x} + c = 0 \]