Victoria's question at Yahoo! Answers regarding a separable first order ODE

[MarkFL]

Here is the question:

General solution of dy/dt=k((y)(b-y))?

B is a constant (in this case, initial temperature)
K is a constant of proportionality
I know you have to use separable variables, but i oeep
Getting stuck. Thanks for the help!

Here is a link to the question:

General solution of dy/dt=k((y)(b-y))? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[MarkFL]

Hello Victoria,

We are given to solve:

\(\displaystyle \frac{dy}{dt}=ky(b-y)\) where $k$ and $b$ are constants.

First, let's separate the variables:

\(\displaystyle \frac{1}{y(b-y)}\,dy=k\,dt\)

Using partial fraction decomposition on the left, and integrating, we obtain:

\(\displaystyle \frac{1}{b}\int\left(\frac{1}{y}-\frac{1}{y-b} \right)\,dy=k\int\,dt\)

\(\displaystyle \ln\left|\frac{y}{y-b} \right|=bkt+C\)

Converting from logarithmic to exponential form, we find:

\(\displaystyle \frac{y}{y-b}=Ce^{bkt}\)

Solving for $y$, we get:

\(\displaystyle y(t)=\frac{b}{1-Ce^{-bkt}}\) where \(\displaystyle 0<C\).

To Victoria and any other guests viewing this topic, I invite and encourage you to post of differential equations problems in our Differential Equations forum.

Best Regards,

Mark.

 

[chisigma]

For completeness sake it should be specified that it exists also the solution y=0 that can't be obtained fon any value of the constant C. In general that happens when an ODE has the form $\displaystyle \frac{d y}{d x} = f(y)$ where is $\displaystyle f(0)=0$ ...


Kind regards


$\chi$ $\sigma$

 

[MarkFL]

Good catch, chisigma!

I failed to mention that during the process of separation of variables, we in fact did lose the trivial solution \(\displaystyle y \equiv 0\).

 

[chisigma]

A good example of the 'traps' into which one can bump with this type of ODE is given in...


http://www.mathhelpboards.com/f17/interesting-ordinary-differential-equation-3684/#post16751

... where some cases in which the initial condition $y(0)=0$ leads to multiple solutions are reported...

Kind regards

$\chi$ $\sigma$