2022.01.21 16:34 TheEpicPineapple [University Differential Equations and Electrical Circuits] Including an additional constant in the solution of a first-order linear ODE
I am learning about RL and RC circuit response right now, and much of that is solving linear 1st order ODEs. The solution is usually easily found by using the Integrating Factor, but I am a little confused about something. As far as I know, 1st order ODEs require only one constant to find a general solution. But it appears that if you want to specify behavior both at t=0 and t=∞, then you can actually use TWO constants. For example:
Initial Condition for current through an inductor: -4.00mA (solved through mesh current by setting the inductor to a short circuit)
Final Condition for current through an inductor: 1.00mA (solved for the same way) Let's just say the time constant is 0.1
The solution for the current i(t) is then:
i(t) = 1 - 5e-10t. By inspection, one can see that the behavior at t=0 (initial state) is satisfied because i(0) = -4 and the behavior at t=∞ (final state) is satisfied because i(∞) = 1
This suggests that if I have a first-order homogeneous linear ODE with constant coefficients
y'(t) + ky = 0
the solution is actually y(t) = Ae-kt + C where C describes the asymptotic behavior at t=∞ and A + C describes the behavior at t=0
If you take the derivative of this equation, the C goes away anyway...so then why did I learn that 1st-order ODEs only require one constant (and that adding another constant is in fact opening me up to the possibility of overdetermination)? Why can I easily add a constant to describe the behavior at ∞ if I so desire without messing up the form of the original equation? I thought there was only supposed to be one constant of integration introduced. Could this be a case of linear superposition that I am kinda being dim about? In short, what is the reason (please not just by inspection) that I can just say "+ C" in the general solution without messing up the original form of the differential equation? Thanks!
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2022.01.21 16:34 shrimpnwine 2022 DCSB OR Progress!
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2022.01.21 16:34 Bonus1Fact [News Shorts] Giants hiring Joe Schoen as new general manager ¦ NY Post on Rumble
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2022.01.21 16:34 Apprehensive-Oil5249 Pavia/ADA Exchange????
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2022.01.21 16:34 ASICmachine Pulling prices from CMC to google sheets but I need more accurate prices (x-post from /r/Cryptocurrency)
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2022.01.21 16:34 TheOnlyDrifter Server "uptime"
Anyone host or rent a server do you known if the server keeps running when no one is online?
Looking at renting one at 4players.com I think was the one was going with
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2022.01.21 16:34 Azami13 D - Fantasy Inspiration (Maria Amanda)
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2022.01.21 16:34 AsterTheArtist [FANART] Found my old ink drawing of some Josuke panels from Diamond is Unbreakable. It looks good when scanned so the materials were just standing there... menacingly
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2022.01.21 16:34 Salt-Contribution-62 What would you ask a six year old?
2022.01.21 16:34 R_Jay101 At what income did you start feeling financially stable?
Obviously, everyones circumstances are different (e.g. living alone compared to living with a partner with 3 kids).
Was there a point where you werent stressing as much about money?
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2022.01.21 16:34 Kind-Dragonfly-2247 Will I survive ME?
Hi! I’m currently a MGT freshie who plans to shift to ME next school year. I just have some questions in mind about the course:
2022.01.21 16:34 Positive_Support8877 College readiness survey (NYC high school students & 10th-12th graders only)
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2022.01.21 16:34 RedMystical As a rock/metalhead, I just recently acquired these vinyls. Am I a fan yet?
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2022.01.21 16:34 jookco Death - Obituary : Louie Anderson, St. Paul-native & Emmy-winning comedian, passed away today. In 1985, @MPR's Beth Friend discussed with Anderson his social work & comedy careers, his comedic inspirations, & his burgeoning stardom. Listen to the segment in the MPR Archive: Click link to read ful...