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

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The discussion revolves around solving the separable first-order ordinary differential equation (ODE) dy/dt = k(y)(b-y), where k and b are constants. The solution process involves separating variables and using partial fraction decomposition, leading to the general solution y(t) = b/(1 - Ce^(-bkt)), with C being a constant. It is noted that the trivial solution y = 0 is also valid but not captured in the general solution due to the separation process. Participants encourage further engagement on differential equations in a dedicated forum. The conversation highlights the complexities and potential pitfalls when solving such ODEs.
MarkFL
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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.
 
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Hello Victoria,

We are given to solve:

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

First, let's separate the variables:

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

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

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

$$\ln\left|\frac{y}{y-b} \right|=bkt+C$$

Converting from logarithmic to exponential form, we find:

$$\frac{y}{y-b}=Ce^{bkt}$$

Solving for $y$, we get:

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

To Victoria and any other guests viewing this topic, I invite and encourage you to post of differential equations problems in our http://www.mathhelpboards.com/f17/ forum.

Best Regards,

Mark.
 
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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 $\chi$ $\sigma$
 
Good catch, chisigma!

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

$\chi$ $\sigma$
 
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