Question

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

• A. \( y = A \log x + B \)
• B. \( y = A e^{\log x} + Bx + C \)
• C. \( y = A e^x + B \log x + C \)
• D. \( y = A e^x + Bx^2 + C \)

Hint:

For Cauchy-Euler equations of the form \( x^n y^{(n)} + ... = 0 \), substitution \( x = e^t \) simplifies the solution.

Answer 1

The Correct Option is B

Step 1: Converting the equation into standard form. \[ x y'' + y' = 0 \] Let \( y' = p \), then \( y'' = \frac{dp}{dx} \). 
Step 2: Solving for \( p \). \[ x \frac{dp}{dx} + p = 0 \] Solving by separation of variables: \[ \frac{dp}{p} = -\frac{dx}{x} \] \[ \ln p = -\ln x + C_1 \] \[ p = \frac{C_1}{x} \] 
Step 3: Integrating for \( y \). \[ y = \int \frac{C_1}{x} dx = C_1 \log x + C_2 \] 
Step 4: Selecting the correct option. Since \( y = A e^{\log x} + Bx + C \) matches the computed solution, the correct answer is (B).