Question

One solution (about $ x = 0 $ ) of the differential equation
$$ x^2 \frac{d^2 y}{dx^2} - 3x \frac{dy}{dx} + 4y = 0 $$ is $ y_1(x) = c_1x^2$ . A second linearly independent solution (with another constant $ c_2 $ ) is:

• A. \( c_2 x^2 \)
• B. \( c_2 x^2 \ln(x) \)
• C. \( \frac{c_2}{x} \)
• D. \( c_2 \ln(x) \)

Hint:

For Euler-Cauchy equations, the second solution is often a logarithmic term multiplied by the first.

Answer 1

The Correct Option is B

For Euler-Cauchy equations, the second solution follows the form:
\[ y_2(x) = y_1(x) \ln(x) \]