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 $\frac{d y}{d x}+2 y \sec ^2 x=2 \sec ^2 x+3 \tan x \cdot \sec ^2 x$...

Let $y=y(x)$ be the solution of the differential equation $\left(x^2+1\right) y^2-2 x y=\left(x^4+2 x^2+1\right) \cos x$. $y(0)=1$. Then $\int_{-2}^3 y(x) \mathrm{d} x$ is:

If a curve $y=y(x)$ passes through the point $\left(1, \frac{\pi}{2}\right)$ and satisfies the differential equation $...

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

Let $y=y_1(x)$ and $y=y_2(x)$ be the solution curves of the differential equation $\frac{d y}{d x}=y+7$ with initial conditions $y_1(0)=y_2(0)=1$ respectively. Then the curves $y=y_1(x)$ and...

Let $x=x(y)$ be the solution of the differential equation $2(y+2) \log _e(y+2) d x+\left(x+4-2 \log _e(y+2)\right) d y=0, y>-1$ with $x\left(e^4-2\right)=1$. Then $x\left(e^9-2\right)$ is equal...