The solution of ${y^5}x + y - x\frac{{dy}}{{dx}} = 0$ is

Which one of the following curves represents the solution of the initial value problem $Dy = 100 - y$,where $y(0) = 50$?

The solution of the differential equation $\cos^2 x \frac{d^2y}{dx^2} = 1$ is

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

$A$ solution curve of the differential equation $(x^2+xy+4x+2y+4) \frac{dy}{dx}-y^2=0, x>0$,passes through the point $(1,3)$. Then the solution curve

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 $x \in R$