Let $f:[1, \infty) \rightarrow[2, \infty)$ be a differentiable function. If $10 \int_1^{x} f(t) dt = 5x f(x) - x^5 - 9$ for all $x \geq 1$,then the value of $f(3)$ is:

Let $y=y(x)$ be the solution of the differential equation $(x+1) y^{\prime}-y=e^{3 x}(x+1)^{2}$,with $y(0)=\frac{1}{3}$. Then,the point $x=-\frac{4}{3}$ for the curve $y = y ( x )$ is

If $x y^{\prime}+y-e^x=0$ and $y(a)=b$, then $\lim _{x \rightarrow 1} y(x)$ is

Solve the following differential equation: $\left(x^2+1\right) \frac{dy}{dx} + 4xy = \frac{1}{x^2+1}$

The solution of the differential equation $x \frac{dy}{dx} + 2y = x^2$ $(x \neq 0)$ with $y(1) = 1$ is

$A$ function $f(x)$ satisfies the condition $f(x) = f'(x) + f''(x) + f'''(x) + \dots \infty$,where $f(x)$ is an indefinitely differentiable function and the dash denotes the order of the derivative. If $f(0) = 1$,then $f(x)$ is: