Suppose $f(x)$ is a differentiable real function such that $f(x) + f'(x) \le 1$ for all $x$ and $f(0)=0$. The largest possible value of $f(1)$ is

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

The general solution of the differential equation $(x^2+2) dy + 2xy dx = e^x(x^2+2) dx$ is

Let $f : R \rightarrow R$ be a differentiable function with $f(0)=1$ and satisfying the equation $f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y)$ for all $x, y \in R$. Then,the value of $\log _e(f(4))$ is:

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:

The solution of the differential equation $\frac{dy}{dx} - 2\frac{y}{x} = x^3$ is: