Learn Laplace Transform Equations with Plainmath

Determine the Laplace Transform ${\displaystyle {(t-1)}^4}$?

As part of trying to solve a differential equation using Laplace transforms, I have the fraction ${\displaystyle \frac{-10s}{(s^2+2)(s^2+1)}}$ which I am trying to perform partial fraction decomposition on so that I can do a inverse Laplace transform. When I try to work out this fraction, I get that ${\displaystyle -10=0(A+B)}$ since ${\displaystyle s^1}$ does not show up in the fraction. How does partial fraction decomposition work for a fraction like this?

According to this example using the Laplace Transform, I am just curious to HOW we go from
${\displaystyle Y\left(s\right)=\frac{32}{s\cdot (s^2+12s+32)}}$
to
${\displaystyle Y\left(s\right)=\frac{32}{s\cdot (s+4)\cdot (s+8)}}$
More specific, I am curious to why we get an 4 and an 8. What formula or equation are being used to get those numbers?I know that the Laplace Transform calculates it for us, however, I want to calculate it "by hand" and show why we get the answer. Any feedback would be gladly appreciated!

What is the Laplace transform of the function ${\displaystyle {\sin }^2\left(\omega t\right)}$

What is the inverse laplace transform of ${\displaystyle F\left(s\right)=\frac{1}{2}\ln \left(\frac{s^{{}^{\left\{2\right\}}}+b^2}{s^{{}^{\left\{2\right\}}}+a^2}\right)}$ with ${\displaystyle a,bϵ\mathbb{R}}$?

calculate the inverse laplace of ${\displaystyle F\left(s\right)=e^{-3s}\frac{1+s}{s^3+2s^2+2s}}$

solve the initial value problem y' +3y =f(t) ,y(0)=1 where f(t) =1 for 0<=t<1 and 0 t>=1

Find the Laplace Transform of cos h 3t cos 2t

Find the laplace form of u(t-2) is

How to solve for the equation y′′′+4y′′+5y′+2y=10cost using laplace transform method given that y(0)=0,y′(0)=0,and y′′(0)=3

I am trying to show this kind of non-linear ${\displaystyle y{}^{⁗}=y^{\prime }y\frac{{}^{″}}{1+x}}$ in normal form. For example here if ${\displaystyle y=e^x\to y^{\left(n\right)}=e^x\to x=-1}$, where ${\displaystyle y^{\left(n\right)}}$ means n-th differential, then ${\displaystyle x=-1}$, too weak idea. When I google with differential or anything like that, most of the material does not look the material that I need. I need to solve different type of problems such as this homework
$$\{\begin{array}{l}{u}^{\prime }=(u+v{)}^2\\ v^{″}=x+u^{\prime }{v}^{\prime }\end{array}$$
I am not requesting you to solve them but I am requesting some material because I find my book quite hard-reading in this section. The earlier chapter begun that something is something, without much further ado really why?, and now the next advanced chapters are referring to the past chapters. The idea is a rush introduction to this topic in an engineering course so I think it explains quite a bit about the pedagogy.

If the coefficients ${\displaystyle A_i}$ of the differential equation are real, then real-valued solutions are generally preferable. Since non-real roots z then come in conjugate pairs, so do their corresponding basis functions ${\displaystyle x^k{e}^{zx}}$, and the desired result is obtained by replacing each pair with their real-valued linear combinations Re(y) and Im(y), where y is one of the pair.

Explicit differential equations
Find all solutions of differential equation
${\displaystyle y^{\prime 2}-(x+y)y^{\prime }+xy=0}$?