List of top Ordinary Differential Equations Questions

Let \( \alpha \) and \( \beta \) be the roots of the indicial equation obtained in the method of finding the Frobenius series solution to the differential equation: \[ 2x^2 \frac{d^2y}{dx^2} - x\frac{dy}{dx} + (1 - x^2) y = 0. \] Which of the following statements is/are TRUE?

The solution \(y(x) = \binom{y_1(x)}{y_2(x)}\) of the initial value problem \[ \frac{dy}{dx} = \begin{pmatrix} -3 & 4 \\ -2 & 3 \end{pmatrix} y, \qquad y(0) = \binom{1}{2} \] is equal to

Consider the following differential equation:

\[ 2x(x-2)^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} + (x-2) y = 0 \]

Which of the following statements is/are TRUE?