Need help with Nonhomogeneous Differentail Equation! | Mathematics | Page 1

sacred fractal Mar 1, 2026, 2:05 PM

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Hey, I’m currently studying for my upcoming Maths for Engineers III exam and need help with that math problem:

Find the solution for that nonhomogeneous differential equation: y‘‘‘-y‘‘-y‘+y = xe^x.
I‘ve already found the homogeneous solution y homo = c1e^x + c2xe^x + c3e^-x. How do I continue?

foggy pendantBOT Mar 1, 2026, 2:05 PM

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steel holly Mar 1, 2026, 6:11 PM

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using undetermined coefficients you would guess an appropriate solution and solve for the required coefficients

sacred fractal Mar 1, 2026, 9:00 PM

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sorry, I'm not really following 🙁

steel holly Mar 1, 2026, 9:01 PM

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how have you dealt with non-homogeneous equations before?

sacred fractal Mar 1, 2026, 9:05 PM

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so we are supposed to make this in the form y non-homo = y homo + y s, where s means 'special'

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I'm sorry, I'm a native German and not used to the English technical words

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are willing to guide me through my dispair? 😅

steel holly Mar 1, 2026, 9:09 PM

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yes but have you come up with the particular solution before? if so, how?

sacred fractal Mar 1, 2026, 9:14 PM

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thank you! no, I haven't and that's where I'm stuck currently

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I tried to differenciate y homo and tried to insert it in my original equation, but this didn't quite work out

steel holly Mar 1, 2026, 9:18 PM

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well have you come up with particular solutions for this type of equations before?

sacred fractal Mar 1, 2026, 9:19 PM

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no not really 😓

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and I dont really understand how to obtain it

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I know you're somewhere supposed differentiate it and insert it somewhere, right?

steel holly Mar 1, 2026, 9:22 PM

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so the idea for this method that we will guess a solution which has some unknown constants in it, and then plug it into the equation

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this page has lots of examples of how to do that:
https://tutorial.math.lamar.edu/Classes/DE/UndeterminedCoefficients.aspx

Differential Equations - Undetermined Coefficients

In this section we introduce the method of undetermined coefficients to find particular solutions to nonhomogeneous differential equation. We work a wide variety of examples illustrating the many guidelines for making the initial guess of the form of the particular solution that is needed for the method.

sacred fractal Mar 1, 2026, 9:25 PM

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so that really the farthest i have come

sacred fractal Mar 1, 2026, 9:26 PM

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steel holly this page has lots of examples of how to do that: https://tutorial.math.lamar.ed...

thank you! I will look into it
can I come back to you if I still dont understand it? 😅

steel holly Mar 1, 2026, 9:26 PM

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sure

sacred fractal Mar 1, 2026, 9:27 PM

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thx

sacred fractal Mar 1, 2026, 10:04 PM

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so uhh

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I'm kinda stuck again...

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did I do smt wrong? 😢

steel holly Mar 1, 2026, 10:10 PM

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you have the right idea for the guess, but you run into the problem that it is part of your homogeneous solution

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if you have that situation, you have to multiply it by the x^s, where s is the number of times e^x showed up as part of the homogenous solution

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why ^3?

sacred fractal Mar 1, 2026, 10:15 PM

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sorry!

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you said as much as e^x shows up in my homo solution, which would be 3 times

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and what do I multiply x^s with?

steel holly Mar 1, 2026, 10:17 PM

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your homogeneous solution has e^x twice and e^-x once

steel holly Mar 1, 2026, 10:17 PM

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sacred fractal

i think you maybe forgot to copy the - sign in your notes here

sacred fractal Mar 1, 2026, 10:17 PM

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yeah, sorry!

sacred fractal Mar 1, 2026, 10:18 PM

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sacred fractal and what do I multiply x^s with?

what about that?

steel holly Mar 1, 2026, 10:19 PM

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your guess for y_p

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so now it is x^s e^x (Ax + B)

sacred fractal Mar 1, 2026, 10:22 PM

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hold on! I found the example on your website where this is discussed XD

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let me read this real quick

sacred fractal Mar 1, 2026, 11:30 PM

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is that correct? it took me almost an hour to write all of that down 😭

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pls say it's correct 👉 👈

steel holly Mar 1, 2026, 11:37 PM

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it appears to be correct

sacred fractal Mar 1, 2026, 11:39 PM

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https://tenor.com/view/happy-dance-dancing-anime-girl-gif-5699276185411794439

Tenor

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holy fucking shit

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tyyyy

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but I have one last question

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why do we guess with (Ax+B) in the case of xe^x / a doubling in solutions, when we subsequently can stick with a simple A and B?

steel holly Mar 1, 2026, 11:44 PM

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i'm not sure what you mean by "a simple A and B"

sacred fractal Mar 1, 2026, 11:53 PM

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in a more simple OED, where the right side is i.e. a Polynomial, you can stick with simple variables like A * p(x) + B * q(x) +..., yet in a non-linear right side like xe^x, you have to use (Ax+B), or am I wrong?

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I'm sorry if I mess up some technical terms, I'm fairly new to this topic - you have to get the gist though XD

steel holly Mar 1, 2026, 11:55 PM

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if the right side is a polynomial, your guess is a polynomial of the same degree

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e.g. right side is x^2 + 3, you guess Ax^2 + Bx + C

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if that polynomial is multiplied by e^x you multiply your guess by e^x to match

sacred fractal Mar 1, 2026, 11:58 PM

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so why did we guess Ax+B for xe^x ?

steel holly Mar 2, 2026, 12:03 AM

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because x is a first order polynomial

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like if the right side was just x we would guess Ax + B, all that changed is that it was multiplied by e^x

sacred fractal Mar 2, 2026, 12:17 AM

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sorry, I still don't really understand your derivation
It might be due to the late hour from where I live

I really have to thank you for your help
I could never have done it without you - so thank you!

#Need help with Nonhomogeneous Differentail Equation!