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

The solution of the equation $\frac{dy}{dx} + 2y \tan x = \sin x$ satisfying $y = 0$ when $x = \frac{\pi}{3}$ is:

At any point on a curve, the slope of the tangent is equal to the sum of the abscissa and the product of the ordinate and abscissa of that point. If the curve passes through $(0, 1)$, then the equation of the curve is

Let the slope of the tangent line to a curve at any point $P(x, y)$ be given by $\frac{xy^2 + y}{x}$. If the curve intersects the line $x + 2y = 4$ at $x = -2$,then the value of $y$,for which the point $(3, y)$ lies on the curve,is ..... .

Let $y^{\prime}(x) + y(x) g^{\prime}(x) = g(x) g^{\prime}(x)$,$y(0) = 0$,$x \in \mathbb{R}$,where $f^{\prime}(x)$ denotes $\frac{d f(x)}{d x}$ and $g(x)$ is a given non-constant differentiable function on $\mathbb{R}$ with $g(0) = g(2) = 0$. Then the value of $y(2)$ is

The solution of the equation $x^2 y - x^3 \frac{dy}{dx} = y^4 \cos x$,where $y(0) = 1$,is