Use a linear approximation to estimate the number 8.07 <mrow class="MJX-TeXAtom-ORD">

Solve the given differential equation. If an initial condition is given, also find the solution that satisfies it.
${\displaystyle (e^x+1)\frac{dy}{dx}=y-y{e}^x}$

Differentiate.
${\displaystyle y=x{\cos }^{-1}x}$

Solve the differential equation $xdy/dx-y=x^2\ln x$

We have a theorem that says: Let $h:I\to \mathbb{R}$ and $g:J\to \mathbb{R}$ be continuous functions, $t_0\in I$ and $y_0\in \text{int(J)}$, hen the differential equation y'(t)=g(y(t))h(t) has a unique solution in some surrounding of $t_0$ if $g(y_0)\ne 0$
How I determine the magnitude of this surrounding? Especially, if I integrate the differential equation and solve it for y(t), am I only able to say: My solution is only unique in the surround of $y_0$ for which g(y(t)) is not zero, is this the condition that determines my surrounding? I mean, I could have determined a solution that gives me zero for some values of t, but is the only solution for my given problem? Am I correct, that in this case, my theorem is not able to say something about the uniqueness of this solution?

Find the general solution of ${\displaystyle y{}^{″}+9y=4\cos 2x}$

Solve initial value problem using laplace transform tables
$y^{\prime }=t+3,y(0)=2$