Recent questions in Separable Differential Equations

Separable Differential Equations
Answered

Dean Summers
2022-07-25

Separable Differential Equations
Answered

Deborah Wyatt
2022-07-23

I am having some trouble with the following separable differential equation

$\frac{dx}{dt}=x(x-1)(x-3)$

with initial condition $x(0)=2$. What is $\underset{t\to \mathrm{\infty}}{lim}x(t)$?

I am having some trouble with the logarithmic laws when solving for $x(t)$

Separable Differential Equations
Answered

Hayley Bernard
2022-07-23

When solving a separable differential equation we move from the original equation

$N(y){\displaystyle \frac{dy}{dx}}=M(x)$

to

$N(y)dy=M(x)dx$

Then we integrate both sides. My question is how precise is the expression $N(y)dy=M(x)dx$ ? is it a formal writing to simplify computation and get quickly into integrating both sides or is it a precise mathematical expression that has a precise mathematical meaning but goes beyond an introductory course on differential equations? Thanks for your help !

Separable Differential Equations
Answered

Paxton Hoffman
2022-07-18

Please guide me:

${y}^{\prime}+ay+b=0$

(a not zero) is supposed to be separable and has solution

$y=c{e}^{-ax}-\frac{b}{a}$

Here is my start to this problem:

$\frac{dy}{dx}+ay=-b$ is as far as I can go with this. How should I go about separating x and

Separable Differential Equations
Answered

suchonosdy
2022-07-17

How would I go about solving the following separable differential equation?

$\frac{dx(t)}{dt}=8-3x$ with $x(0)=4$?

My solution thus far is the following:

$\int \frac{dx(t)}{8-3x}=\int dt+C$

$\Rightarrow \text{ln}|8-3x(t)|=-3(t+C)$

Now using the fact that $x(0)=4$ we can solve for $C=-\frac{1}{3}\text{ln}|4|$. From this, I would seem to get

$|8-3x(t)|={e}^{-3t-\frac{1}{3}\text{ln}4}$

but something seems to go wrong. Am I making a mistake somewhere and how could I solve this entirely?