The general solution of the differential equation $\frac{d^2 y}{d x^2}+8 \frac{d y}{d x}+16 y=0$ is

Let $f$ be a differentiable function such that $2(x+2)^2 f(x) - 3(x+2)^2 = 10 \int_0^x (t+2) f(t) dt$ for $x \geq 0$. Then $f(2)$ is equal to . . . . . .

The solution of the differential equation $x^2 \frac{dy}{dx} - xy = 1$ is...

Let $f$ be a differentiable function such that $x^2 f(x) - x = 4 \int_0^x t f(t) dt$ and $f(1) = \frac{2}{3}$. Then $18 f(3)$ is equal to $......$.

Let $x = x(y)$ be the solution of the differential equation $2(y + 2) \log_e(y + 2) dx + (x + 4 - 2 \log_e(y + 2)) dy = 0$,$y > -1$ with $x(e^4 - 2) = 1$. Then $x(e^9 - 2)$ is equal to

The solution of $\frac{dy}{dx} + 2y \tan x = \sin x$ is