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

The solution of the differential equation $x \cos x \frac{dy}{dx} + (x \sin x + \cos x) y = 1$ is

The solution of $\frac{dy}{dx} + \frac{1}{3}y = 1$ is

Let $y=y(x)$ be the solution of the differential equation $x \log _e x \frac{d y}{d x}+y=x^2 \log _e x, (x > 1)$. If $y(2)=2$,then $y(e)$ is equal to

Suppose $y=y(x)$ is the solution curve to the differential equation $\frac{dy}{dx}-y=2-e^{-x}$ such that $\lim_{x \rightarrow \infty} y(x)$ is finite. If $a$ and $b$ are respectively the $x$- and $y$-intercepts of the tangent to the curve at $x=0$,then the value of $a-4b$ is equal to:

The integrating factor of the differential equation $(1 - y^2)\frac{dx}{dy} + yx = ay$ for $(-1 < y < 1)$ is