The integrating factor of the linear differential equation $\frac{dy}{dx} = \frac{1}{4x + 3y}$ is

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

Let the slope of the tangent to a curve $y=f(x)$ at $(x, y)$ be given by $2 \tan x(\cos x-y)$. If the curve passes through the point $(\frac{\pi}{4}, 0)$,then the value of $\int_{0}^{\pi / 2} y \, dx$ is equal to

Let $y = y(x)$ be the solution of the differential equation $(x^2 + 1)^2 \frac{dy}{dx} + 2x(x^2 + 1)y = 1$ such that $y(0) = 0$. If $\sqrt{a} y(1) = \frac{\pi}{32}$,then the value of $a$ is

If $\frac{dy}{dx} + \frac{3}{\cos^2 x} y = \frac{1}{\cos^2 x}$,$x \in \left( -\frac{\pi}{3}, \frac{\pi}{3} \right)$ and $y\left( \frac{\pi}{4} \right) = \frac{4}{3}$,then $y\left( -\frac{\pi}{4} \right)$ equals

If the curve $y=y(x)$ is the solution of the differential equation $2(x^{2}+x^{5/4}) dy - y(x+x^{1/4}) dx = 2x^{9/4} dx, x > 0$ which passes through the point $(1, 1-\frac{4}{3} \log_{e} 2)$,then the value of $y(16)$ is equal to :