Question

The general solution of $(x^2 D^2 - xD)y = 0$ is

• A. $y = c_1 + c_2 e^x$
• B. $y = c_1 + c_2 x$
• C. $y = c_1 + c_2 x^2$
• D. $y = c_1 x + c_2 x^2$

Hint:

An alternative quick method is to assume a solution of the form $y = x^m$.
Substituting this directly gives: $m(m-1) - m = 0 \Rightarrow m^2 - 2m = 0$, leading immediately to the roots $m = 0, 2$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question asks for the general solution of a second-order, linear homogeneous differential equation with variable coefficients, specifically in the Cauchy-Euler form.

Step 2: Key Formula or Approach:
The given equation is:
\[ x^2 \frac{d^2y}{dx^2} - x \frac{dy}{dx} = 0 \] To solve a Cauchy-Euler equation, we use the transformation:
\[ x = e^z \quad \Rightarrow \quad z = \ln x \] This substitution converts the variable-coefficient ODE into a constant-coefficient ODE using the operator relations:
\[ x D = D_z \] \[ x^2 D^2 = D_z(D_z - 1) \] Where $D_z = \frac{d}{dz}$.

Step 3: Detailed Explanation:

• Substitute the operator expressions into the differential equation:
\[ [D_z(D_z - 1) - D_z]y = 0 \] \[ [D_z^2 - D_z - D_z]y = 0 \] \[ [D_z^2 - 2D_z]y = 0 \]
• Write the auxiliary equation by replacing the operator $D_z$ with the variable $m$:
\[ m^2 - 2m = 0 \]
• Factor the equation to find the roots:
\[ m(m - 2) = 0 \] This gives the roots:
$m_1 = 0$
$m_2 = 2$

• Write the solution in terms of the intermediate variable $z$:
\[ y(z) = c_1 e^{0z} + c_2 e^{2z} = c_1 + c_2 e^{2z} \]
• Convert the solution back to the original variable $x$ using $e^z = x$:
\[ y(x) = c_1 + c_2 (e^z)^2 = c_1 + c_2 x^2 \]

Step 4: Final Answer:
The general solution is $y = c_1 + c_2 x^2$, which corresponds to option (C).