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 _____