Let $S=(0,2 \pi)-\left\{\frac{\pi}{2}, \frac{3 \pi}{4}, \frac{3 \pi}{2}, \frac{7 \pi}{4}\right\}$. Let $y=y(x), x \in S$, be the solution curve of the differential equation $...

Let $y=y(x)$ be the solution of the differential equation $x\left(1-x^2\right) \frac{d y}{d x}+\left(3 x^2 y-y-4 x^3\right)=0, x>1$, with $y(2)=-2$. Then $y(3)$ is equal to

Let the solution curve $y=y(x)$ of the differential equation, $\left[\frac{x}{\sqrt{x^2-y^2}}+e^{\frac{y}{x}}\right] x \frac{d y}{d x}=x+\left[\frac{x}{\sqrt{x^2-y^2}}+e^{\frac{y}{x}}\right] y$ pass through the points $(1,0)$ and $(2 \alpha, \alpha), \alpha>0$....

Let the solution curve of the differential equation $x \frac{d y}{d x}-y=\sqrt{y^2+16 x^2}, y(1)=3$ be $y=y(x)$. Then $y(2)$ is equal to i

If $\frac{d y}{d x}+\frac{2^{x-y}\left(2^y-1\right)}{2^x-1}=0, x, y>0, y(1)=1$, then $y(2)$ is equal to :

If the solution curve $y=y(x)$ of the differential equation $y^2 d x+\left(x^2-x y+y^2\right) d y=0$, which passes through the point $(1,1)$ and intersects the line...

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 :

If $x=x(y)$ is the solution of the differential equation $y \frac{d x}{d y}=2 x+y^3(y+1) e^y, x(1)=0$; then $x(e)$ is equal to :

Let the tangent at any point P on a curve passing through the points $(1,1)$ and $\left(\frac{1}{10}, 100\right)$, intersect positive $x$-axis and $y$-axis at the...