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

Solve by using the Laplace Transform Method?
${\displaystyle y^{\prime }+y=6e^{t^2},y\left(0\right)=3}$

Solve differential equation$x{y}^{\prime }-2x^{2}y=e^{(}{x}^2)$

Suppose u is a twice continuously differentiable function with linear growth,
$\underset{x\to \mathrm{\infty }}{lim}{u}^{\prime }(x)-\frac{1}{g(x)}u(x)=0$
and g is a Lipschitz continuous function with Lipschitz constant $L<1$.
Consider the first order linear homogenous differential equation
$y^{\prime }(x)-\frac{1}{g(x)}y(x)=0.$
The general solution is
$y(x)=c\exp (\int \frac{1}{g(x)}dx)$
for constant $c\in \mathbb{R}$. In any solution with linear growth, $\underset{x\to \mathrm{\infty }}{lim}y(x)=0$
Is it possible to conclude that $\underset{x\to \mathrm{\infty }}{lim}u(x)=0$?

Write an equivalent first-order differential equation and initial condition for y.
${\displaystyle y=-4+{\int }_1^x(4t-y\left(t\right))dt}$
What is the equivalent first-order differential equation?
What is the initial condition?

Solve differential equation ${\displaystyle \frac{dy}{dx}={(x+y+1)}^2-{(x+y-1)}^2}$

The marginal productive output of workers in a small manufacturing firm is given by $MP=8L-L^2+20$ where L is the number of workers hired by the firm. Rewrite the equation above as a first order differential equation and find the general solution for the total productive output function P(L).

Solve the first order linear differential equation
${\displaystyle \frac{dy}{dx}+xy=x,y\left(0\right)=-6}$