PYQs Based Test - 12 : Higher order LDE

If roots of the auxiliary equation of $\frac{d^{2}y}{d{x}^2}+a\frac{dy}{dx}+by=0$ are real and equal, the general solution of the differential equation is

The general solution of the differential equation $\frac{d^{4}y}{d{x}^4}-2\frac{d^{3}y}{d{x}^3}+2\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+y=0$ is

If y = f(x) is solution of $\frac{d^{2}y}{d{x}^2}=0$ with the boundary conditions y = 5 at x = 0, and $\frac{dy}{dx}=2$ at x = 10, f(15) = _____

If the characteristic equation of the differential equation $\frac{d^{2}y}{d{x}^2}+2\alpha \frac{dy}{dx}+y=0$ has two equal roots, then the values of α are

Consider the following second order linear differential equation

$\frac{d^{2}y}{d{x}^2}=-12x^2+24x-20$

The Boundary conditions are: at x=0, y=5 and x=2, y=21

The value of y at x=1 is?