List of practice Questions

For the differential equation given below, which one of the following options is correct? \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \quad 0 \leq x \leq 1, 0 \leq y \leq 1 \]

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). \[ x^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} + y = 0 \] \[ y(x = 1) = 0, \quad \frac{dy}{dx}(x = 1) = 1 \]

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). \[ x^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} + y = 0 \] \[ y(x = 1) = 0, \quad \frac{dy}{dx}(x = 1) = 1 \]