Question

The solution of the differential equation \(x \frac{d^{2} y}{dx^{2}} + \frac{dy}{dx} = 0\) is

• A. \(y = c_{1} \sin x + c_{2}\)
• B. \(y = c_{1} \cos x + c_{2}\)
• C. \(y = c_{1} \log x + c_{2}\)
• D. \(y = c_{1} e^{x} + c_{2} e^{-x}\)

Hint:

Notice that the equation can be written using the product rule:
$x \frac{d^2 y}{dx^2} + \frac{dy}{dx} = \frac{d}{dx}\left( x \frac{dy}{dx} \right) = 0$.
Integrating once gives $x \frac{dy}{dx} = c_1$, and integrating a second time leads directly to $y = c_1 \ln x + c_2$.

Answer 1

The Correct Option is C