Question

If \( y(x) \) is a solution of \[ x^2 y'' - 4 x y' + 6 y = 0, y(-1) = 1, y'(-1) = 0, \] then the value of \( y(2) \) is \(\underline{\hspace{2cm}}\).

Hint:

For Cauchy-Euler equations, use the transformation \( y(x) = x^r \) to reduce the equation to a quadratic characteristic equation.

Answer 1

Correct Answer: 28

The given second-order linear differential equation is of the form: \[ x^2 y'' - 4 x y' + 6 y = 0. \] We solve this equation using standard methods for solving Cauchy-Euler equations. After solving the equation, we substitute the initial conditions \( y(-1) = 1 \) and \( y'(-1) = 0 \) to find the specific solution. Finally, we calculate \( y(2) \) to get: \[ y(2) = 28. \] Thus, the value of \( y(2) \) is \( 28 \).