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...

If $y=y(x)$ is the solution of the differential equation $\frac{d y}{d x}+\frac{4 x}{\left(x^2-1\right)} y=\frac{x+2}{\left(x^2-1\right)^{5 / 2}}, x>1$ such that $\mathrm{y}(2)=\frac{2}{9} \log _{\mathrm{e}}(2+\sqrt{3})$ and $...

Let $f(x)=\frac{x}{\left(1+x^n\right)^{\frac{1}{n}}}, x \in R-\{-1\}, n \in N, n>2$. If $f^n(x)=$ (fofof ..... upto $n$ times) ( $x$ ), then $...

For the system of equations $$ \begin{aligned} & x+y+z=6 \\ & x+2 y+\alpha z=10 \end{aligned} $$ $x+3 y+5 z=\beta$, which one of the following is...

Let $f(x)=x-1$ and $g(x)=e^x$ for $x \in \mathbb{R}$. If $\frac{d y}{d x}=\left(e^{-2 \sqrt{x}} g(f(f(x)))-\frac{y}{\sqrt{x}}\right), y(0)=0$, then $y(1)$ is

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}+2 y \sec ^2 x=2 \sec ^2 x+3 \tan x \cdot \sec ^2 x$...