Series solution of xy^{' '}+2y^{'}-xy=0 I get r(r+1)=0, (r+1)(r+2)c_1=0 and c

Find $\frac{dw}{dt}$ using the appropriate Chain Rule. Evaluate $\frac{dw}{dt}$ at the given value of t. Function: $w=x\sin y,x=e^t,y=\pi -t$ Value: $t=0$

Solve the differential equation by variation of parameters
${\displaystyle y+3y^{\prime }+2y=\frac{1}{1+e^x}}$

Convert unit measures for a problem from zill book
I have the solution, but I don't understand it, I don't understand the part which implies the conversion of unit measures. I know that I can ${\displaystyle g=32\cdot \frac{ft}{s^2}}$, but then what? So my questions are:
1. how to compute (convert unit measures) ${\displaystyle \frac{4pound}{32\cdot \frac{ft}{s^2}}}$ to become ${\displaystyle \frac{1}{8}}$?
pound is a unit measure for weight, ft is for distance (like meter), s is for time. So, how they (in image) simplify that raport - division?
2. Then, for the next step: I know that ${\displaystyle {\omega }^2=\frac{k}{m}}$, with ${\displaystyle m=\frac{1}{8}}$ slug and ${\displaystyle k=16\frac{lb}{ft}}$. But what about this division:
${\displaystyle \frac{k}{m}=\frac{16\frac{lb}{ft}}{\frac{1}{8}slug}=128\frac{lb\cdot slug}{ft}}$
And again, how can I make this division? lb and slug are units of measure for weight and ft is for distance. How can I reduce them?

Second order differential inequality and comparison theorem.
If ${\displaystyle x:[0,1]\Rightarrow \mathbb{R},x\in C^{\mathrm{\infty }}}$ that satisfies
$$\{\begin{array}{l}\ddot{x}\le -2x\\ x(0)=x(1)=0\end{array}$$
is it true that ${\displaystyle x\ge 0\text{on}[0,1]}$?
I observed that if ${\displaystyle x\ge 0}$, then ${\displaystyle \ddot{x}\le 0}$, so x is concave. However I could not find out any other properties. I think if there exists a solution of the following differential equation
$$\{\begin{array}{l}\ddot{x}=-2x\\ x(0)=x(1)=0\end{array}$$
then some kind of comparative theorem can be used. However, there is no non-trivial solution for this.
If the above proposition is incorrect, could you give me a counterexample?

Solve the equation:
${\displaystyle x(2y+1)dx=y(x^2-3x+2)dy}$

Prove instability using Lyapunov function
${\displaystyle x^{\prime }=x^3+xy}$
${\displaystyle y^{\prime }=-y+y^2+xy-x^3}$

I have to solve the following Cauchy's problem:
$\{\begin{array}{rl}& x^2{x}^{\prime }={\sin }^2(x^3-3t)\\ & x(0)=1\end{array}.$