Convert the differential equation u''-u'-2u=e^{-5t} into a system of first

Solve differential equation$x^2{y}^{\prime }+xy=x^2+y^2$

Using Laplave Transform, evaluate the integro-differential equation
$y^{″}(x)+9y(x)=40e^x;y(0)=5,y^{\prime }(0)=-2$

The Laplace transform L ${\displaystyle \left\{e^{-t^2}\right\}}$ exists, but without finding it solve the initial-value problem ${\displaystyle y{}^{″}+9y=3e^{-t^2},y\left(0\right)=0,y^{\prime }\left(0\right)=0}$

We have the following differential equation
${\displaystyle \cup {}^{″}=1+{\left(u^{\prime }\right)}^2}$
i found that the general solution of this equation is
${\displaystyle u=d\text{cosh}((x-\frac{b}{d})}$
where b and d are constats
Please how we found this general solution?

My question is to find the solutions to the following
$\frac{df(x)}{dx}=f^{-1}(x)$
where $f^{-1}(x)$ refers to the inverse of the function f. The domain really isn't important, though I am interested in either (-inf, inf) or (0, inf), so if any solutions are known for more restricted domains then they are welcome.
I cannot find any material relating to this type of question in any of my calculus and differential equations textbooks and references; it seems quite unorthodox. Any material which covers this type of diff equation would be wlecome

Solve the first-order differential equations:
${\displaystyle (x^2+1)\frac{dy}{dx}=xy}$

Problem 1.7 in G.Teschl ODE and Dynamical Systems asks me to transform the following differential equation into autonomous first-order system:
$\ddot{x}=t\sin (\dot{x})+x$
Transforming the ODE to a system is in this case easy, but whats the usual technique to transform it to an AUTONOMOUS system?