Solve for G.S./P.S. for the following differential equations using the

Solve differential equation ${\displaystyle x\frac{dy}{dx}-2y=x^3{\sin }^2\left(x\right)}$

Solve differential equation $dy/dx-12x^{3}y=x^3$

Dont

Are the following are linear equation?
${\displaystyle 1.\frac{dP}{dt}+2tP=P+4t-2}$
${\displaystyle 2.\frac{dy}{dx}=y^2-3y}$

Solve the equation separable, linear, bernoulli, or homogenous
${\displaystyle 1.\frac{dy}{dx}=\frac{x^4+4x{y}^2}{2x^3+x^{2}y+y^3}}$
${\displaystyle 2.\left(e^{-y}\cos \left(x\right)\right)y^{\prime }=x^4+6x^2{y}^3}$
${\displaystyle 3.y^{\prime }=\frac{y+y{x}^3}{x+x^2}\cos \left(\frac{x^2}{y^2}\right)}$

Write an equivalent​ first-order differential equation and initial condition for y.
${\displaystyle y=-1+{\int }_0^x(2t-5y\left(t\right))dt}$
What is the equivalent first-order differential equation?
${\displaystyle y^{\prime }=?}$
What is the initial condition?
${\displaystyle y(y^{\prime }=?)=?}$

What method would you use to solve:
$(1+x^2)\frac{\mathrm{d}y}{\mathrm{d}x}=1+y^2;y(2)=3$
I am asking this because I only know two methods of solving the DEs - separation of variables and integrating factor. Since the separation of variables does not work here, I tried integrating factor, however, I don't know what to do with the y2, because for the IF to work I need to get y on its own $\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)$)
What method do I use to solve this?