Solve differential equation x'(t)= \frac{\sin (t)}{x^2(t)+1}

Look at the Chain Rule formula with a visual explanation of how it works:
${\displaystyle {\left(h\left(x\right)\right)}^{\prime }={\left(g\left(f\left(x\right)\right)\right)}^{\prime }=g^{\prime }\left(f\left(x\right)\right)\cdot f^{\prime }\left(x\right)}$
${\displaystyle g\left(f\left(x\right)\right)}$-outside function
${\displaystyle \left(f\left(x\right)\right)}$-inside function
${\displaystyle g^{\prime }\left(f\left(x\right)\right)}$-derivative of the otside at f(x)
${\displaystyle f^{\prime }\left(x\right)}$-derivative of the inside
Now, explain what happens in regards to calculating the derivative, if f(x) is also a composite function.

Solve the differential equation $xdy/dx-y=x^2\ln x$

For a given differential equation
$x{y}^{\prime }+2y\log y-4x^{2}y=0;$
$y(1)=1$
I want to use the substitution $v=\log y$. Which implies that
$v^{\prime }=\frac{dv}{dy}\frac{dy}{dx}=\frac{1}{y}{y}^{\prime }$
Hence:
$y^{\prime }=y{v}^{\prime }$
That being said, solving the equation I try as follows:
$xy{v}^{\prime }+2yv-4x^{2}y=0$
$x{v}^{\prime }+2v=4x^2$
$v^{\prime }+\frac{2}{x}v=4x$
$v^{\prime }=4x-\frac{2}{x}v$
How can I use seperation of variables here? Or how can I solve this thing ?

Solve and classify equation $(k+1)dy+kydx=0$

I am curious if there are any examples of functions that are solutions to second order differential equations, that also parametrize an algebraic curve.
I am aware that the Weierstrass $\mathrm{\wp }$ - Elliptic function satisfies a differential equation. We can then interpret this Differential Equation as an algebraic equation, with solutions found on elliptic curves. However this differential equations is of the first order.
So, are there periodic function(s) $F(x)$ that satisfy a second order differential equation, such that we can say these parametrize an algebraic curve?
Could a Bessel function be one such solution?

Solve differential equation $(1+y^2+x{y}^2)dx+(x^{2}y+y+2xy)dy=0$

The Runge Kutta method is known as:
a. An analytical method of solving first order differential equations.
b. An accurate method of solving first order differential equations.
c. A numberical method of solving first order differential equations.
d. A complex method of solving first order differential equations.