Let the tangent at any point $P(x, y)$ on a curve passing through the points $(1, 1)$ and $(\frac{1}{10}, 100)$ intersect the positive $x$-axis and $y$-axis at the points $A$ and $B$ respectively. If $PA: PB = 1: k$ and $y = y(x)$ is the solution of the differential equation $e^{\frac{dy}{dx}} = 2x + 1$ with $y(0) = 2$,then $4y(1) - 5 \log_e 3$ is equal to:

The solution of the differential equation $\frac{d^2 y}{d x^2}+y=0$ is

$A$ solution of $y = 2x\left( \frac{dy}{dx} \right) + x^2\left( \frac{dy}{dx} \right)^4$ is

Verify that the given function $y - \cos y = x$ is a solution of the differential equation $(y \sin y + \cos y + x) y' = y$.

Let $f:(-1,1) \rightarrow R$ be a differentiable function satisfying $(f^{\prime}(x))^4 = 16(f(x))^2$ for all $x \in (-1,1)$ and $f(0)=0$. The number of such functions is:

Let $b$ be a nonzero real number. Suppose $f: R \rightarrow R$ is a differentiable function such that $f(0)=1$. If the derivative $f^{\prime}$ of $f$ satisfies the equation $f^{\prime}(x) = \frac{f(x)}{b^2+x^2}$ for all $x \in R$,then which of the following statements is/are $TRUE$?
$(A)$ If $b>0$,then $f$ is an increasing function
$(B)$ If $b < 0$,then $f$ is a decreasing function
$(C)$ $f(x)f(-x)=1$ for all $x \in R$
$(D)$ $f(x)-f(-x)=0$ for all $x \in R$