Help solving a 1st order ODE

I'm trying to solve analytically the following *simple* first order differential equation:

$\displaystyle {dx \over dt} = \frac{(1-x)}{\tau} - Ux$

for $\displaystyle \tau$ and $\displaystyle U$ being constants. In principle, this differential equation is *linear*, right? However, the only thing I could think at the moment is to try to write it differently to approximate to the following expression:

$\displaystyle {dx(t) \over dt} +p(t)x = q(t)$

that has a general solution of the form:

$\displaystyle x(t) = \frac{\int e^{\int p(t)dt} q(t)dt + C}{e^{\int p(t) dt}}$

For that, what I did so far:

1. **Reorganize** the formula by multiplying everything by $\displaystyle \tau$:

$\displaystyle \tau {dx \over dt} = (1-x)- Ux \tau $

2. **Group** the terms involving the unknown:

$\displaystyle \tau {dx \over dt} = 1 -x(1 + U \tau) $

3. **Reorganizing** we can write as:

$\displaystyle \tau {dx \over dt} + x(1 + U \tau) = 1$

4. **Dividing** by $\displaystyle \tau$

$\displaystyle {dx \over dt} + x\frac{(1 + U \tau)}{\tau} = 1/\tau$

This gives me $\displaystyle p(x) = (1+U)/\tau$, and $\displaystyle q(x) = 1/\tau$ which are constants, and not functions.

Is this the right way to proceed or should I try another method?

Thanks for your help

Re: Help solving a 1st order ODE

Yes, that is a linear equation but it is also "separable" and I think that is an easier approach.

$\displaystyle \frac{dx}{dt}= \frac{1- x}{\tau}- Ux= \frac{1- x- \tau Ux}{\tau}$

Separate the variables: $\displaystyle \tau \dfrac{dx}{1- (1+\tau U)x}= \dfrac{dt}{\tau}$

and each side can be integrated (on the left let $\displaystyle y= 1- (1+ \tau U)x$).

Re: Help solving a 1st order ODE

Wov! Thanks!

I first tried to do it with by separation of variables, but I wasn't able to separate the equation into 2 expressions:

From your *separated* expression:

$\displaystyle \frac{dx}{1-(1+\tau U)x}= \frac{dt}{\tau}$

Let be $\displaystyle y = 1-(1+\tau U)x$,

then $\displaystyle dy= -(1+U\tau)dx$,

both expressions that we substitute in the differential equation:

$\displaystyle \frac{dy}{y (-U\tau - 1)}= \frac{dt}{\tau}$

Reorganize the negative symbol:

$\displaystyle \frac{dy}{y (U\tau + 1)}= -\frac{dt}{\tau}$

and moving the constants to the left side:

$\displaystyle \frac{dy}{y}= -\frac{ (U\tau + 1)}{\tau} dt$

then solving for y:

$\displaystyle \log(y) = -\frac{ (U\tau + 1)}{\tau} t $

and applying exponentiation to both sides...

$\displaystyle y= \exp{(-\frac{ (U\tau + 1) t}{\tau})}$

I'm not very sure this is the right answer....

Re: Help solving a 1st order ODE

I really don't know if I am allowed to do all this but I gave new found LaTeX a work out anyway.

$\displaystyle \begin{align*}\frac{dx}{dt}=&\frac{1-x}{r}-Ux\\\frac{dx}{dt}=&\frac{(1-x)-Urx}{r}\\\frac{dt}{dx}=&\frac{r}{1-x-Urx}\\\frac{dt}{dx}=&\frac{r}{1-x(1+Ur)}\\t=&r\times \int \frac{1}{1-x(1+Ur)}dx\\t=&r\times \frac{ln(1-x(1+Ur))}{-(Ur+1)}+k\;\;\text{where k is a constant}\\t=&\frac{-r}{Ur+1}\times ln(1-(Ur+1)x)+k\\\frac{(t-k)(Ur+1)}{-r}=&ln(1-(Ur+1)x)\\e^\frac{(k-t)(Ur+1)}{r}=&1-(Ur+1)x\\\frac{e^{\frac{(k-t)(Ur+1)}{r}}-1}{-(Ur+1)}=&x\\x=&\frac{1-e^{\frac{(k-t)(Ur+1)}{r}}}{Ur+1}\end{align*}$

Am I allowed to do this?

Re: Help solving a 1st order ODE

Hey Melody2!!!

Unless I'm doing something wrong, your solution

$\displaystyle x(t) = \frac{1- \exp(\frac{(K-t) (U \tau + 1)}{\tau}))}{(U \tau + 1)}$

is not correct.

I've just tested whether the solution of that my CAS provide give the same return values as your equation with the same parameters (K=1, U=0.8 and tau=2) for some values(for example for t=1 and t=5), and they're different. Strange enough, the steady-state conditions match.

See details here

my solution above doesn't work either.

Re: Help solving a 1st order ODE

Quote:

Originally Posted by

**jguzman** Wov! Thanks!

I first tried to do it with by separation of variables, but I wasn't able to separate the equation into 2 expressions:

From your *separated* expression:

$\displaystyle \frac{dx}{1-(1+\tau U)x}= \frac{dt}{\tau}$

Let be $\displaystyle y = 1-(1+\tau U)x$,

then $\displaystyle dy= -(1+U\tau)dx$,

both expressions that we substitute in the differential equation:

$\displaystyle \frac{dy}{y (-U\tau - 1)}= \frac{dt}{\tau}$

Reorganize the negative symbol:

$\displaystyle \frac{dy}{y (U\tau + 1)}= -\frac{dt}{\tau}$

and moving the constants to the left side:

$\displaystyle \frac{dy}{y}= -\frac{ (U\tau + 1)}{\tau} dt$

then solving for y:

$\displaystyle \log(y) = -\frac{ (U\tau + 1)}{\tau} t $

and applying exponentiation to both sides...

$\displaystyle y= \exp{(-\frac{ (U\tau + 1) t}{\tau})}$

I'm not very sure this is the right answer....

This is almost correct. The solution is

$\displaystyle x = A e^{- \left ( 1 + \tau U)t/ \tau} \right )$

(You forgot to put it in terms of x and also the constant of integration.)

-Dan

Re: Help solving a 1st order ODE

Hi

Quote:

Originally Posted by

**topsquark** This is almost correct. The solution is

$\displaystyle x = A e^{- \left ( 1 + \tau U)t/ \tau} \right )$

-Dan

Actually, I'm not very sure about that, for 2 reasons:

1) Your equation differs from the one that results from the symbolic integration computed with software (see here)

2) More importantly, note that the solution of the differential equation in steady-state depends on $\displaystyle U$ and $\displaystyle \tau$, since

$\displaystyle {dx \over dt}= 0$,

then $\displaystyle \frac{(1-x)}{\tau}-Ux =0$,

operating and organizing we have

$\displaystyle (1-x) = Ux\tau$,

$\displaystyle 1 - x - Ux\tau = 0$,

$\displaystyle x + Ux\tau = 1$,

solving for x:

$\displaystyle x= \frac{1}{(1+U\tau)}$,

when your equation tends to infinity, the value of x tends to zero, because $\displaystyle e^{-\infty}=0$

Now I know that the solution is like this:

$\displaystyle x(t)=\frac{{\left(U \tau + e^{\left(U t +\frac{t}{\tau}\right)}\right)} e^{\left(-U t\right)}}{U \tau e^{\frac{t}{\tau}} + e^{\frac{t}{\tau}}}$

But do not know how to get there... or to get a more simple expression

Re: Help solving a 1st order ODE

Perhaps this will help: The A in my equation is a constant that depends on U and tau.

-Dan

Re: Help solving a 1st order ODE

Quote:

Originally Posted by

**topsquark** This is almost correct. The solution is

$\displaystyle x = A e^{- \left ( 1 + \tau U)t/ \tau} \right )$

(You forgot to put it in terms of x and also the constant of integration.)

-Dan

I think it will help me to try to solve it again:

From my (correct?) expression:

Quote:

Originally Posted by

**jguzman**

$\displaystyle \frac{dy}{y}= -\frac{ (U\tau + 1)}{\tau} dt$

We apply integration to the previous expression:

$\displaystyle \int \frac{dy}{y}= -\frac{ (U\tau + 1)}{\tau} \int dt,$

and solve that integral, we obtain:

$\displaystyle \log{y}= -\frac{ (U\tau + 1)}{\tau} t + A,$

being $\displaystyle A$ the integration constant. Now we have to apply exponential to both sides. In this the step I'm not very sure if the exponential of the left hand side is correct (I assumed that exponential of a sum is not the same as the sum of the exponentials).

$\displaystyle y= \exp \Big(-\frac{ (U\tau + 1)t}{\tau} + A\Big),$

Since we know that $\displaystyle y = 1 - (1+U\tau)x$ we can write:

$\displaystyle 1 - (1+U\tau)x = \exp \Big(-\frac{ (U\tau + 1)t}{\tau} + A\Big),$

reorganizing everything:

$\displaystyle - (1+U\tau)x = \exp \Big(-\frac{ (U\tau + 1)t}{\tau} + A\Big) - 1 ,$

$\displaystyle (1+U\tau)x = 1 - \exp \Big(-\frac{ (U\tau + 1)t}{\tau} + A\Big) ,$

which results in this monster:

$\displaystyle x(t) = \frac{1 - \exp \Big(-\frac{ (U\tau + 1)t}{\tau} + A\Big)}{(1+U\tau)} $

I missed something...

Re: Help solving a 1st order ODE

I believe that this last answer is exactly the same as my answer

Your A equals my (Ur+1)k/r They are both just constants.

Re: Help solving a 1st order ODE

Quote:

Originally Posted by

**jguzman** Hey Melody2!!!

I've just tested whether the solution of that my CAS provide give the same return values as your equation with the same parameters (K=1, U=0.8 and tau=2) for some values(for example for t=1 and t=5), and they're different. Strange enough, the steady-state conditions match.

Hi jguzman,

My knowledge has holes in it and I actually didn't understand what you meant by steady-state. I can see that as t gets bigger f(t) appears to be or approach the same constant. Maybe 0.4 I assume that that is what you meant.

I am also interested in the Sage code that you used. Is that program an internet freeby?

Melody.

Re: Help solving a 1st order ODE

Hi Melody!

Quote:

Originally Posted by

**Melody2** Hi jguzman,

My knowledge has holes in it and I actually didn't understand what you meant by steady-state. I can see that as t gets bigger f(t) appears to be or approach the same constant. Maybe 0.4 I assume that that is what you meant.

That's exactly what I meant by steady-state. The differential equation has an steady-state if the function x(t) does not change anymore with the independent variable (in my case t). In that case, the rate of change of x(t) is zero, therefore the differential equation is zero. The value that I obtained in the steady state dependended on both $\displaystyle \tau$ and $\displaystyle U$. The relationship is:

$\displaystyle x(\infty) = \frac{1}{1+U\tau}$

For the numerical computations I performed, I set $\displaystyle U=0.8$ and $\displaystyle \tau=2$, so we have an steady-state of 0.38

Both yours and mysolution have the same steady state, so I guess they only differ in the way the constants are defined. The definition of the constants is important in my case because I'm trying to model a biophysical system.

Quote:

Originally Posted by

**Melody2** I am also interested in the Sage code that you used. Is that program an internet freeby?

Melody.

Yes, Sage is free. In principle you can use it in your computer, via a cloud or in Internet via the link I sent you (this allows you to share pieces of code with colleagues, as I did here).

I will send you a link via a private message and some examples, ok?

Re: Help solving a 1st order ODE

Thanks very much for your thoughtful answer

and

I would like the link.

Thanks

Melody.