Recent content by Robaj

Thanks both, it seems trivial in hindsight!

I guess the question is: how can I get to ##y(t)=2(e-1)e^{-t}## when ##g(t)=0##?

Thanks for pointing out my mistake. How can I solve for ##y(t)## when ##g(t)=0## if the second integral always vanishes? If ##g(t)=0## then $$ \begin{align} \frac{y'}{y} &= -1\\ \ln y &= -\int_1^t 0e^s ds +c = 0 \end{align} $$ Perhaps I'm confusing the integration limits with the initial...

Thanks, this gives me the result for ##0\le t\le 1##, although now I'm not sure about the case for ##t\gt 1##.

Non-homegenous first order ODE so start with an integrating factor ##\mu## $$\mu=\textrm{exp}\left(\int a dt\right)=e^t.$$ Then rewrite the original equation as $$\frac{d}{dt}\mu y = \mu g(t).$$ Using definite integrals and splitting the integration across the two cases, $$\begin{align}...

Ah I understand. This is very interesting! Thanks for both explanations.

Ah, I see. Understanding how the author got from (1) to (2) is really what I'm after! But I appreciate your help. Your second post has cleared up some confusion I had about differentials.

I see what you're saying. But surely the derivation goes in one direction only, so we should be able to go forward at each point based only on what we've derived previously. It doesn't make sense to me to go backwards!

Thanks for your patience - I'm struggling to join the dots. I want to get from the first equation to the second but your substitution doesn't match the denominator of that first equation. Just to be clear, I'm looking at $$\begin{equation}...

Thanks for your reply. I understand the differentiation and substitution you've used, but not how they help me get from the first equation to the second equation in the top post. Could you clarify? I may have misunderstood how to get from ## uf(u)\frac{du}{dt} ## to ## \frac{d f(u)}{dt}. ##

Hi, I've been following a derivation of relativistic kinetic energy. I've seen other ways to get the end result but I'm interested in finding out where I've gone wrong here: I'm struggling with integrating by parts. The author goes from...

Ok, I'll leave it as it is. Thanks a lot.

Homework Statement From Prussing and Conway (Q1.11): derive an expression for the eccentricity e in terms of the initial speed v, radius r, and flight path angle x (they use gamma). Homework Equations (1) h^2 = mu*a*(1-e^2) [a is semimajor axis, mu is gravitational parameter] (2) h =...