Why does the method of separable equations work in differential equations? In differential equations, the idea of multiplying by an infinitesimal, dx, is used in the method of separable equations. My confusion is in why this works as I've heard many times in the past that you shouldn't multiply and cancel out infinitesimals like that. I understand that in the setting of nonstandard analysis that there may not be something wrong with this, but in the usual setting is there a more rigorous interpretation of this?

1. A body falls from rest against resistance proportional to the square root of the speed at any instant. If the body attains speed V1 and V2 feet per second, after 1 and 2 seconds in motion, respectively, find an expression for the limiting velocity.

The Laplace transform of the product of two functions is the product of the Laplace transforms of each given function. True or False

The Laplace transform of ${\displaystyle u(t-2)}$ is
(a) ${\displaystyle \frac{1}{s}+2}$
(b) ${\displaystyle \frac{1}{s}-2}$
(c) ${\displaystyle e^2\frac{s}{s}\left(d\right)\frac{e^{-2s}}{s}}$??

1 degree on celsius scale is equal to
A) $\frac{9}{5}$ degree on fahrenheit scale
B) $\frac{5}{9}$ degree on fahrenheit scale
C) 1 degree on fahrenheit scale
D) 5 degree on fahrenheit scale

The Laplace transform of $t{e}^t$ is A. $\frac{s}{(s+1{)}^2}$ B. $\frac{1}{(s-1{)}^2}$ C. $\frac{s}{(s+1{)}^2}$ D. $\frac{s}{(s-1)}$

What is the Laplace transform of ${\displaystyle t\cos t}$ into the s domain?

Find the general solution of the given differential equation:
$y^{″}-<!--\; -\; -->2y^{\prime }+y=0$

The rate at which a body cools is proportional to the difference in 
temperature between the body and its surroundings. If a body in air 
at 0℃ will cool from 200℃ 𝑡𝑜 100℃ in 40 minutes, how many more 
minutes will it take the body to cool from 100℃ 𝑡𝑜 50℃ ?

A body falls from rest against a resistance proportional to the velocity at any instant. If the limiting velocity is 60fps and the body attains half that velocity in 1 second, find the initial velocity.

What's the correct way to go about computing the Inverse Laplace transform of this?
$\frac{-<!--\; -\; -->2s+1}{(s^2+2s+5)}$
I Completed the square on the bottom but what do you do now?
$\frac{-<!--\; -\; -->2s+1}{(s+1{)}^2+4}$

How to find inverse Laplace transform of the following function?
$X(s)=\frac{s}{s^4+1}$
I tried to use the definition: $f(t)={\mathcal{L}}^{-<!--\; -\; -->1}\{F(s)\}=\frac{1}{2\mathrm{\pi <!--\; \pi \; -->}i}\underset{T\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}{\int <!--\; \int \; -->}_{\mathrm{\gamma <!--\; \gamma \; -->}-<!--\; -\; -->iT}^{\mathrm{\gamma <!--\; \gamma \; -->}+iT}{e}^{st}F(s)ds$or the partial fraction expansion but I have not achieved results.

How do i find the lapalace transorm of this intergral using the convolution theorem? ${\int <!--\; \int \; -->}_0^t{e}^{-<!--\; -\; -->x}\cos <!--\; \; -->xdx$

How can I solve this differential equation? : ${\displaystyle xy dx-(x^2+1) dy=0}$

Find the inverse Laplace transform of $\frac{s^2-<!--\; -\; -->4s-<!--\; -\; -->4}{s^4+8s^2+16}$

inverse laplace transform - with symbolic variables:
$F(s)=\frac{2s^2+(a-<!--\; -\; -->6b)s+a^2-<!--\; -\; -->4ab}{(s^2-<!--\; -\; -->a^2)(s-<!--\; -\; -->2b)}$
My steps:
$F(s)=\frac{2s^2+(a-<!--\; -\; -->6b)s+a^2-<!--\; -\; -->4ab}{(s+a)(s-<!--\; -\; -->a)(s-<!--\; -\; -->2b)}$
$=\frac{A}{s+a}+\frac{B}{s-<!--\; -\; -->a}+\frac{C}{s-<!--\; -\; -->2b}+K$
$K=0$
$A=F(s)\ast <!--\; \ast \; -->(s+a)$