Find the equation by applying the Laplace transform. displaystyle{y}^{{{left({4}right)}}}-{y}= sin{{h}}{t} y(0)=y'(0)=y"(0)=0 y'''(0)=1

Which of the following is true about Lagranges

Solve: ${\displaystyle \frac{dy}{dx}-\frac{dx}{dy}=\frac{y}{x}-\frac{x}{y}}$

I solved the differential equation
$\frac{dy}{dx}=\frac{x}{x^2+1}$
to get the general solution
$y=\frac{ln|x+1|+c}{2}$
Im given the initial condition
$y{y}^{\prime }-2e^x=0,y(0)=3$
but I dont know what to do with it

Find the general solution of the second order non-homogeneous linear equation:
$y^{″}-7y^{\prime }+12y=10\sin t+12t+5$

I have a first order linear differential equation (a variation on a draining mixing tank problem) with many constants, and want to separate variables to solve it.
$\frac{dy}{dt}=k_1+k_2\frac{y}{k_3+k_{4}t}$
y is the amount of mass in the tank at time t, and for simplicity, I've reduced various terms to constants, $k_1$ through $k_4$.
Separation of variables is made difficult by $k_1$, and I've considered an integrating factor, but think I might be missing something simple.

Having trouble solving this separable differential equation
I am having some trouble with the following separable differential equation
$\frac{dx}{dt}=x(x-1)(x-3)$
with initial condition $x(0)=2$. What is ${\displaystyle \underset{t\to \mathrm{\infty }}{lim}x(t)}$?
I am having some trouble with the logarithmic laws when solving for $x(t)$

The Laplace transform ${\displaystyle L\left\{e^{-t^2}\right\}}$ exists, but without finding it solve the initial-value problem ${\displaystyle y{}^{″}+y=e^{-t^2},y\left(0\right)=0,y^{\prime }\left(0\right)=0}$