Solve System of Differential Equations with Expert Help

Find the general solution y(x) of the following second order linear ODEs:
${\displaystyle y{}^{″}-4y^{\prime }+5y=0}$

Which of the following is not a solution of ${\displaystyle y{}^{″}+4y=0}$
(A) ${\displaystyle 4\cos \left(2x\right)}$
(B) ${\displaystyle 5\sin \left(2x\right)}$
(C) ${\displaystyle \sin \left(2x\right)\cos \left(2x\right)}$
(D) ${\displaystyle 4\cos \left(2x\right)+5\sin \left(2x\right)}$

Detemine whether the equation given is a solution to the corresponding differential equation.
${\displaystyle y=C_1{e}^{-1.25t}+c_2{e}^{-5.5t}-16;}$
${\displaystyle y{}^{″}+\frac{27}{4y^{\prime }}+\frac{55}{8y}=-110}$

Find the particular solution to the initial value problem
${\displaystyle y{}^{″}-2y^{\prime }+y=0,y\left(0\right)=1,y^{\prime }\left(0\right)=-1}$

Find the general solution of the following differential equation (where primes denote derivatives with respect to x):
${\displaystyle y{}^{″}+8y^{\prime }+16=0}$

Solve the equation
${\displaystyle y{}^{″}-4y=2+3x-x^2}$

Solve the equation:
${\displaystyle y{}^{″}+2y^{\prime }+2y=0,y\left(\frac{\pi }{4}\right)=2,y^{\prime }\left(\frac{\pi }{4}\right)=-2}$

Find the solution of the following initial value problem:
${\displaystyle x{}^{″}\left(t\right)-7x^{\prime }\left(t\right)+10x\left(t\right)=0,x\left(0\right)=2,x^{\prime }\left(0\right)=3}$

Identify the auxiliary equation of the following differential equation along with its roots and the two fundamental solutions.
${\displaystyle y{}^{″}+6y^{\prime }+8y=0}$

Solve the given differential equation by undetermined coefficients.
${\displaystyle y{}^{″}+4y=4\sin \left(2x\right)}$

Find the particular solution of the following linear second-order inhomogeneous differential equation
${\displaystyle \frac{d^{2}y\left(t\right)}{{dt}^2}-2\frac{dy\left(t\right)}{dt}+4y\left(t\right)=3t-4}$

Solve the following differential equation
${\displaystyle y{}^{″}+4y=\cos x\sin x}$

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