Question

Let \( y_1(x) \) and \( y_2(x) \) be two linearly independent solutions of the differential equation: \[ x^2 \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0, \quad x>0. \] Let \( W(y_1, y_2)(x) \) denote the Wronskian of \( y_1(x) \) and \( y_2(x) \) at \( x \). If \( W(y_1, y_2)(1) = 1 \), then \( W(y_1, y_2)(2) \) is equal to ________.

Hint:

For Cauchy-Euler equations, the Wronskian of two solutions is given by \( W(y_1, y_2)(x) = Cx \), where \( C \) is determined from the initial condition.

Answer 1

Correct Answer: 4

The given differential equation is: \[ x^2 \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0 \] This is a Cauchy-Euler equation, and for this type of equation, the Wronskian \( W(y_1, y_2)(x) \) satisfies the following property: \[ \frac{dW}{dx} = \frac{1}{x} W \] The solution to this equation is: \[ W(y_1, y_2)(x) = Cx \] where \( C \) is a constant. We are given that \( W(y_1, y_2)(1) = 1 \), so: \[ C \times 1 = 1 \quad \Rightarrow \quad C = 1 \] Thus, the Wronskian is: \[ W(y_1, y_2)(x) = x \] Now, we can calculate \( W(y_1, y_2)(2) \): \[ W(y_1, y_2)(2) = 2 \] Thus, the value of \( W(y_1, y_2)(2) \) is: \[ \boxed{2} \]