If $y=y(x)$ and $\frac{2+\sin x}{y+1}\left(\frac{d y}{d x}\right)=-\cos x$,with $y(0)=1$,then $y\left(\frac{\pi}{2}\right)$ is equal to

If the slope of the tangent at a point $(x, y)$ on a curve is $\frac{y-4}{x-3}$ and the curve passes through $(4, 3)$,then the point where it cuts the line $y=x$ is

Let $x=x(t)$ and $y=y(t)$ be solutions of the differential equations $\frac{dx}{dt}+ax=0$ and $\frac{dy}{dt}+by=0$ respectively,where $a, b \in R$. Given that $x(0)=2$,$y(0)=1$,and $3y(1)=2x(1)$,the value of $t$ for which $x(t)=y(t)$ is:

If the solution of $\frac{dy}{dx} - y \log_{e} 0.5 = 0$,$y(0) = 1$,and $y(x) \rightarrow k$,as $x \rightarrow \infty$ then $k =$

The solution of the differential equation $(1 + x^2)(1 + y)dy + (1 + x)(1 + y^2)dx = 0$ is

The general solution of the differential equation $(3x-4y)(dx-3dy)+(6dx-4dy)=0$ is