If the order and degree of the differential equation corresponding to the family of curves $y^2=4a(x+a)$ (where $a$ is a parameter) are $m$ and $n$ respectively, then $m+n^2=$

Let $f(x)$ be a positive function such that the area bounded by $y=f(x), y=0$ from $x=0$ to $x=a>0$ is $e^{-a}+4a^2+a-1$. Then the differential equation,whose general solution is $y=c_1 f(x)+c_2$,where $c_1$ and $c_2$ are arbitrary constants,is :

The order of the differential equation whose solution is $y=a \cos x+b \sin x+c e^{-x}$ is

Let the tangent to a curve at point $(x, y)$ intersect the $Y$-axis at point $P$. $A$ line drawn through point $P$ is perpendicular to this tangent and passes through the point $(1, 0)$. The differential equation of the curve is:

If the family of curves $y = a e^{4x} + b e^{-x}$, where $a, b$ are arbitrary constants, represents the general solution of the differential equation $f(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}) = 0$, then find $\frac{df}{dx}$.

The differential equation whose solution is $(x - h)^2 + (y - k)^2 = a^2$ is ($a$ is a constant).