Recent content by FranzS

Yes, sorry, I replied too much in a hurry during lunch break. :smile: I'll think about your last post thank you

The sketch you've drawn would probably not work like the system I described. The real situation is:

A mechanism acts to compress the rod/piston. During compression, the pressure inside the two chambers (whose volumes change, i.e. volume of the rod side increases and volume of the piston side decreases, and also the sum of the two decreases overally) is ideally equal because of one-way valves...

Ok then, as I said it doesn't matter, but since you are committed to the problem I'll disclose its exact formulation: $$ \begin{align} & P_1'(t)= - P_1(t) \cdot \ \frac{V_1'(t) + c_0 F(t)}{V_1(t)} \\[0.5em] & F(t) = 2 \sqrt{x(t) \big( 1-x(t) \big)} \\[0.5em] & x(t) = 1 - \frac{P_2(t)}{P_1(t)}...

Thanks for your interest. However, I have no freedom on any parameter, they are all predetermined. I see no use in explaining all the calculations, just rest assured that ##F(0)=0## (this is "big ##F##") because it's a factor that determines gas flow through an orifice between two chambers that...

It's not a choice, the constants are dictated by the real problem being analysed. Anyway, I found out that the quadratic power series isn't much better than the linear one. The reason is indeed the steep behaviour of ##F(t)=\sqrt{(...)}## at ##t=0##, i.e. ##F'(t \to 0) \to \infty##, which any...

Sorry, it's my fault. I didn't specify that for ##t=0## the constants combine such that ##x(t=0)=0##.

Thanks again, but this is exactly my original problem: ##g'(f,t)## diverges for##t=0##.

Thanks a lot, but I'm actually interested in a closed form (approximate) solution.

As you may imagine, this differential equation describes the evolution in time of a certain quantity ##f(t)##, and only the initial state of the system is known, namely ##f(0)## as well as all other terms for ##t=0##.

Oh well, it's pretty intricated. I'll write it by steps: $$ \begin{align} & f'(t)= - f(t) \cdot \ \frac{V_1'(t) + c_0 F(t)}{V_1(t)} \\[0.5em] & F(t) = 2 \sqrt{x(t) \big( 1-x(t) \big)} \\[0.5em] & x(t) = 1 - \frac{h(t)}{f(t)} \\[0.5em] & h(t) = \frac{c_1 c_2 - f(t) V_1(t)}{V_2(t)} \\[0.5em] &...

Hi PF, I'm trying to find an approximate solution of a differential equation that can't be solved in exact form. The differential equation is of the form: $$ f'(t)=g(f(t),t) $$ I want to find the approximate solution in terms of a power series: $$ f(t) \approx f(0) + f'(0) \cdot t +...

Yes, this is what I thought. Just compare the effects of static forces with dynamic impacts

It was a backward explanation to me, so to speak. In @fresh_42 specific example it was clear that the power series diverges for ##x \geq 1##. Sorry

This is already a very good explanation to me, thank you