The differential equation for which $ax + by = 1$ is the general solution is:

The degree of the differential equation obtained from the equation $(y - a)^2 = 4(x - b)$ [where $a$ and $b$ are arbitrary constants] is

Form the differential equation representing the family of curves $y=a \sin (x+b),$ where $a$ and $b$ are arbitrary constants.

Statement $I$: The differential equation corresponding to the family of circles having their centres on $Y$-axis and fixed radius $k$ is $(x^2-k^2)(\frac{dy}{dx})^2+x^2=0$.
Statement $II$: The differential equation corresponding to the family of circles passing through the origin and having their centres on $X$-axis is $x^2-y^2+2xy \frac{dy}{dx}=0$.
Which of the above statements is (are) true?

If $a, b, c, d$ are arbitrary constants,then the corresponding differential equation for $y = a e^x + b e^{-x} + c \cos x + d \sin x$ is

If the differential equation having $y=Ae^x+B \sin x$ as its general solution is $f(x) \frac{d^2 y}{d x^2}+g(x) \frac{d y}{d x}+h(x) y=0$, then $f(x)+g(x)+h(x)=$