Question

The general solution of
$$ \left(x\frac{dy}{dx} - y\right)\sin\frac{y}{x} = x^3 e^x $$ 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: The provided differential equation is: \[ \left( x \frac{dy}{dx} - y \right) \sin \frac{y}{x} = x^3 e^x \] This can be rearranged to: \[ x \frac{dy}{dx} - y = \frac{x^3 e^x}{\sin \frac{y}{x}} \]

Step 2: Solve the equation using a suitable technique. Given its linear form, the integrating factor method is applicable. Following simplification, integrate both sides with respect to \(x\). 

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