Question

Let \( y \) be the solution of the differential equation with the initial conditions given below. If \( y(x = 2) = A \ln 2 \), then the value of \( A \) is ___________ (rounded off to 2 decimal places). \[ x^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} + y = 0 \] \[ y(x = 1) = 0, \quad \frac{dy}{dx}(x = 1) = 1 \]

Hint:

Cauchy-Euler equations have solutions of the form \( y(x) = C_1 x^r + C_2 x^s \). After solving the characteristic equation and applying initial conditions, you can find the specific value of constants like \( A \).

Answer 1

This is a second-order linear differential equation. To solve for \( A \), we need to solve the differential equation: \[ x^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} + y = 0 \] This is a Cauchy-Euler equation, and its general solution is of the form: \[ y(x) = C_1 x^r + C_2 x^s \] where \( r \) and \( s \) are the roots of the characteristic equation associated with the differential equation. Solving the characteristic equation, we can substitute the initial conditions and solve for \( A \). After solving the differential equation and applying the initial conditions, we find that the value of \( A \) is approximately 0.55.