The solution of the differential equation $\frac{dy}{dx} - y \tan x = e^x \sec x$ is

The solution of $\frac{dy}{dx} + y = e^{-x}, y(0) = 0$ is

The solution of $(1+x^2) \frac{dy}{dx} + 2xy - 4x^2 = 0$ is:

Let $y=y(x)$ be a solution of the differential equation $(x \cos x) dy + (xy \sin x + y \cos x - 1) dx = 0$,$0 < x < \frac{\pi}{2}$. If $\frac{\pi}{3} y(\frac{\pi}{3}) = \sqrt{3}$,then $|\frac{\pi}{6} y''(\frac{\pi}{6}) + 2 y'(\frac{\pi}{6})|$ is equal to $.........$.

For $x \in R$,let the function $y(x)$ be the solution of the differential equation $\frac{dy}{dx} + 12y = \cos \left(\frac{\pi}{12} x\right)$ with $y(0) = 0$. Then,which of the following statements is/are $TRUE$?

Find the equation of a curve passing through the origin,given that the slope of the tangent to the curve at any point $(x, y)$ is equal to the sum of the coordinates of the point.