Integration of a First order for physical application

[crevoise]

Hello everyone.

I wish to get the solution to the following:

x'(t) = [A*exp(B*t)-C]^(m)

I can get the plotted solution by Matlab, but I wish to find the analytic solution by myself.
Does anyone has some hints to help me in this?

Thanks a lot for your help

/Crevoise

 

[Ackbach]

The RHS is independent of $x$. You can integrate both sides directly, although it's not a pretty integral. You've got yourself a hypergeometric function in there.

 

[chisigma]

The RHS is independent of $x$. You can integrate both sides directly, although it's not a pretty integral. You've got yourself a hypergeometric function in there.

With simple steps You obtain...

$\displaystyle f(t)= (A\ e^{B\ t}-C)^{m}= \{-C\ (1-\frac{A}{C}\ e^{B\ t})\}^{m}= (-1)^{m}\ C^{m}\ \sum_{n=0}^{m} (-1)^{n}\ \binom{m}{n}\ (\frac{A}{C})^{n}\ e^{n\ B\ t}$ (1)

... and (1) can be integrated 'term by term'...

Kind regards

$\chi$ $\sigma$

 

[crevoise]

Thanks a lot for your two answers, really helpful!
I should have thought about the binomial theorem...
Thanks again

 

[blue_raver22]

Hello Crevoise,

The integration of a first-order equation is a fundamental process in physical applications. In order to find the analytic solution to the equation x'(t) = [A*exp(B*t)-C]^(m), there are a few steps you can follow.

First, you can start by separating the variables on both sides of the equation. This means putting all the terms with x on one side and all the terms with t on the other side. In this case, you will have x'(t) = [A*exp(B*t)-C]^(m) on one side and dx on the other side.

Next, you can integrate both sides of the equation. This will give you an equation in the form of x(t) = f(t) + C, where C is the constant of integration. In order to find the value of C, you can use the initial condition given in the problem.

To solve the integral on the left side, you can use the substitution method or integration by parts. For the integral on the right side, you can use the power rule or the substitution method.

Once you have solved the integral, you will have the analytic solution for x(t). You can then use this solution to plot the graph or use it in further calculations.

I hope this helps. Good luck with finding the analytic solution!