List of top Ordinary Differential Equations Questions

The initial value problem \[ \frac{dy}{dx} = \cos(xy), \quad x \in \mathbb{R}, \quad y(0) = y_0, \] where \( y_0 \) is a real constant, has
If the ordinary differential equation \[ x^2 \frac{d^2 \phi}{dx^2} + x \frac{d\phi}{dx} + x^2 \phi = 0, \quad x>0 \] has a solution of the form \( \phi(x) = x^r \sum_{n=0}^{\infty} a_n x^n \), where \( a_n \)'s are constants and \( a_0 \neq 0 \), then the value of \( r^2 + 1 \) is __________.
The Bessel functions \( J_\alpha(x), x>0, \alpha \in \mathbb{R} \) satisfy \[ J_{\alpha-1}(x) + J_{\alpha+1}(x) = \frac{2\alpha}{x} J_\alpha(x). \] Then, the value of \( \left( \pi J_{\frac{3}{2}}(\pi) \right)^2 \) is __________.

The partial differential equation \[ 7 \frac{\partial^2 u}{\partial x^2} + 16 \frac{\partial^2 u}{\partial x \partial y} + 4 \frac{\partial^2 u}{\partial y^2} = 0 \] is transformed to \[ A \frac{\partial^2 u}{\partial \xi^2} + B \frac{\partial^2 u}{\partial \xi \partial \eta} + C \frac{\partial^2 u}{\partial \eta^2} = 0, \] using \( \xi = y - 2x \) and \( \eta = 7y - 2x \). Then, the value of \( \frac{1}{123} (B^2 - 4AC) \) is _____

If \( y = \sum_{k=0}^{\infty} a_k x^k \), \( (a_0 \neq 0) \) is the power series solution of the differential equation
\[ \frac{d^2y}{dx^2} - 24x^2y = 0, \text{ then } \frac{a_4}{a_0} = \text{\(\underline{\hspace{2cm}}\).} \]
Let \( y(t) \) be the solution of the initial value problem \[ \frac{d^2 y}{dt^2} + a \frac{dy}{dt} + b y = f(t), a > 0, b > 0, a \neq b, a^2 - 4b = 0, \] with initial conditions \( y(0) = 0 \), \( \frac{dy}{dt}(0) = 0 \), obtained by the method of Laplace transform. Then
The critical point of the differential equation \[ \frac{d^2y}{dt^2} + 2 \alpha \frac{dy}{dt} + \beta^2 y = 0, \alpha > 0, \, \beta > 0, \] is a
Let \( y(x) \) be the solution of the following initial value problem \[ x^2 \frac{d^2y}{dx^2} - 4x \frac{dy}{dx} + 6y = 0, x > 0, \] \[ y(2) = 0, \frac{dy}{dx}(2) = 4. \] Then \( y(4) = \(\underline{\hspace{1cm}}\) \).
The eigenvalues of the boundary value problem \[ \frac{d^2y}{dx^2} + \lambda y = 0, x \in (0, \pi), \lambda > 0, \] \[ y(0) = 0, y(\pi) - \frac{dy}{dx}(\pi) = 0 \] are given by: