If $\frac{d^2y}{dx^2} + \sin x = 0$,then the solution of the differential equation is ...... .

The solution of the differential equation $\frac{dy}{dx} = \frac{\sin y + e^x}{\ln y - x \cos y}$ 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 $x \in R$

The curve amongst the family of curves represented by the differential equation,$(x^2 - y^2)dx + 2xy\, dy = 0$ which passes through $(1, 1)$,is

Let $y=y(x)$ be the solution of the differential equation $(x+y+2)^2 dx=dy$,$y(0)=-2$. Let the maximum and minimum values of the function $y=y(x)$ in $\left[0, \frac{\pi}{3}\right]$ be $\alpha$ and $\beta$,respectively. If $(3\alpha+\pi)^2+\beta^2=\gamma+\delta\sqrt{3}$,where $\gamma, \delta \in Z$,then $\gamma+\delta$ equals:

The solution of the differential equation $e^{-x}(y+1) dy + (\cos^2 x - \sin 2x) y dx = 0$ at $x=0, y=1$ is