Differential Equations - Engineering Mathematics - Aerospace Engineering Previous Year Questions

The partial differential equation \(\frac{\partial^2 u}{\partial x^2} + 4\frac{\partial^2 u}{\partial x \partial y} + 2\frac{\partial^2 u}{\partial y^2} = 0\) is ____.

Consider the ordinary differential equation: \(\frac{1}{2} \frac{dy}{dx} + \frac{y}{x} = 1\). If \(y = 2/3\) at \(x = 1\), then the value of \(y\) at \(x = 3\) is ____ (rounded off to the nearest integer).

Consider the differential equation with the initial conditions given below. If y(x) is the solution of the equation, the value of the slope, \( \frac{dy}{dx} \), at \( x = \ln(2) \) is __________ (rounded off to three decimal places).
\( \frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = 0 \) with \( y|_{x=0} = 0 \) and \( \frac{dy}{dx}\big|_{x=0} = 1 \)