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

Solve differential equation $y^{\prime }(t)=-3y+9$, $y(0)=4$

Identify the surface whose equation is given.
$\rho =\sin \theta \sin \varphi$

Solve the following first-order nonlinear ordinary differential equation:
${\displaystyle y^{\prime }\left(x\right)+\log \left(y\left(x\right)\right)=-x-1}$

Solve differential equation $(6x+1)y^{2}dy/dx+3x^2+2y^3=0$, y(0)=1

For every differentiable function $f:\mathbb{R}\to \mathbb{R}$, there is a function $g:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ such that $g(f(x),f^{\prime }(x))=0$ for every $x$ and for every differentiable function $h:\mathbb{R}\to \mathbb{R}$ holds that
being true that for every $x\in \mathbb{R}$, $g(h(x),h^{\prime }(x))=0$ and $h(0)=f(0)$ implies that $h(x)=f(x)$ for every $x\in \mathbb{R}$.
i.e every differentiable function $f$ is a solution to some first order differential equation that has translation symmetry.

Solve differential equation $dy/dx-(cotx)y=si{n}^{3}x$

Most systems can be modeled using first order system just like a temperature control system of an incubator.
Question 1: How can we know that a certain system can be modeled using first order differential equation?
${\displaystyle a_1\frac{dy}{dt}+a_{0}y=F\left(t\right)\left(1\right)}$
A first order differential equation in Equation (1) may be written as:
${\displaystyle \tau \frac{dy}{dt}+y=kF\left(t\right)\left(2\right)}$
Question 2: If a step input
${\displaystyle F\left(t\right)=\{\begin{array}{cc}0& \text{if}t<0\\ A& \text{if}t>0\end{array}\left(3\right)}$
of the system has an output response of a
${\displaystyle y\left(t\right)=kA(1-e^{\frac{-t}{\tau }}\left(4\right)}$
then the system can be modeled as first differential equation where
${\displaystyle y_{max}=kA\left(5\right)}$
Is this correct?
Given a step input and the graph of the step response of the system. Am i correct if I say that if the output of the system behaves similar to equation (4), then I can model the system using first differential equation?