The differential equation of all lines perpendicular to the line $5x + 2y + 7 = 0$ is

The differential equation of the family of circles passing through the points $(0,2)$ and $(0,-2)$ 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 $\frac{df}{dx}$.

If the differential equation obtained by eliminating $A$ and $B$ from $y = (\sin^{-1} x)^2 + A \cos^{-1} x + B$ is $(a - x^2) y'' - x y' = b$,then $\frac{b + a}{b - a} =$

The differential equation whose general solution is given by $y = (c_1 \cos(x + c_2)) - (c_3 e^{(-x + c_4)}) + (c_5 \sin x)$,where $c_1, c_2, c_3, c_4, c_5$ are arbitrary constants,is

$y=A e^x+B e^{-2 x}$ satisfies which of the following differential equations?