Question

Consider the differential equation \( x^2 \frac{d^2y}{dx^2} = 6y \). The general solution of the above equation is

Hint:

Recognize the \(x^2y'', xy', y\) pattern as a Cauchy-Euler equation. The substitution \(y=x^m\) quickly converts it into an algebraic auxiliary equation, which is easy to solve.

Answer 1

Step 1: Identify the ODE type.
$x^2 y" - 6y = 0$. This is a Cauchy-Euler equation.
Step 2: Substitute $y = x^m$.
Then $y' = mx^{m-1}$ and $y" = m(m-1)x^{m-2}$. Plugging in: $x^2[m(m-1)x^{m-2}] - 6x^m = 0 \implies x^m[m^2 - m - 6] = 0$.
Step 3: Solve Auxiliary Equation.
$m^2 - m - 6 = 0 \implies (m-3)(m+2) = 0$. Roots $m_1 = 3, m_2 = -2$. General solution: $y = c_1 x^3 + c_2 x^{-2} = ax^3 + b/x^2$.