The general solution of the differential equation of all circles having center at $A(-1, 2)$ is $ . . . . . . $.

The order of the differential equation whose general solution is given by $y = (C_1 + C_2) \sin (x + C_3) - C_4 e^{x + C_5}$ is (where $C_1, C_2, C_3, C_4, C_5$ are arbitrary constants).

For each of the exercises given below,verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
$y=e^{x}(a \cos x+b \sin x) \quad: \frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+2 y=0$

If $\alpha$ and $\beta$ are respectively the order and degree of the differential equation for which $a x^2+b y^2=1$ is the general solution, then the eccentricity of the ellipse $\alpha x^2+\beta y^2=1$ is

The differential equation of the family of lines having $x$-intercept $a$ and $y$-intercept $b$ is

The differential equation satisfied by $y = X \sin(6t + 5) + Y \cos(6t + 5)$ is (where $X$ and $Y$ are constants).