Formula to find y_p1 and y_p2 in ODEs like: y'" - ay" + ay' - y = e^x (x² + bx) + cx²

I made this formula because this type of ODE requires a big resolution

For instance, when looking up the quadratic formula online, I most often come across the following solution:

[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
]

However, in my country, we are taught to identify (b^2 - 4ac) as the discriminant (I hope that's the correct English translation). Before substituting it into the formula, we first check whether it is negative (no real solutions), positive (two real solutions), or equal to zero (one real solution).

This is just one of many differences I've posted so far. Essentially, mathematics is the same everywhere, but different countries seem to have their own approaches to teaching it. These seemingly minor differences have made me wonder...

Hello, I'm a first year college student, so I might not know all of the concepts needed to understand the answer. Today I've discovered something relatively simple, but very interesting, and would like to know if there is already something documenting it.

So we all know that numbers divisible by 9 have their sums of digits divisible by 9

e. g.

4 941 → 4 + 9 + 4 + 1 = 18

137 682 → 1 + 3 + 7 + 6 + 8 + 2 = 27

When I was first learning how to multiply I remember noticing that 9 • n (where 0 < n < 11) all have their sums exactly equal to 9.

11 • 9 equals 99, and this is where I thought this correlation ended. However, today I've realised that it doesn't. 12 • 9 is 108 and the sum goes back to 9 again. It does so until 21. There I thought it just didn't stay constant for two-digit numbers where the last digit was 1, but no. The correlation continues only after 23. It goes on until 31, and starts again at 34.

So the sum of digits of any two-digit n times 9 is equal to 9 only if n's first digit is larger than the second (excluding 0 as the second number).

34 • 9 = 306

59 • 9 = 531

78 • 9 = 702

And so on

After that I wanted to check how it looks like for bigger numbers. And after some test I concluded that the sum of digits in n • 9 is always equal to 9 if n has it's digits in ascending order, counting from left to right (e. g. 1478, 689, 3469). Moreover I've noticed that any digit of n can be repeated infinitely, as long as it meets the previous requirements (e. g. 1444...444578), and the repeated digit isn't the last one. The number 0 can be repeated at the end of n, but that's pretty obvious.

So that's what I've observed. I find the topic very interesting and would be thankful if any of You could direct me to some sort of thesis discussing it, if one exists.

Thanks again for reading

xoxo

Guys, I looked at this problem for 2 hours, but it just seems impossible. The wording is also confusing: I don't really understand what the question means with "exactly ONE point". Anyone have an idea on how to solve it?

Consider the two functions:

f(x) = x² + 1

g(x) = mx + 2

where m is a real number.

Find a value of m such that the two graphs intersect at exactly ONE point, even though neither graph is tangent to the other in the usual way.

Can you find such an m and prove that it works?

Regarding my last post's comments, this is a different question now but same principle.

In a simulation, basically:

You start with a number, let's use 100.

You have a 50% chance of rolling the color green or red.

If you hit GREEN your number goes up by 50%.

If you hit RED your number goes down by 5%.

Also, if the number ever goes <1, the sequence stops.

THE QUESTION:

Given infinite sequences of infinite random rolls, which sequence is more likely to get:

1 - A sequence where it eventually stops, due to a large enough red sequence

2 - A sequence with no reds at all

3 - They have the same probability

I’m trying to remember 9th grade algebra so I can teach my sister.

Above is what I tried. Is it right? If not, what should I do?

I figure that I probably want to get rid of the fraction, but am unclear how to do so.

Does any one use this estimations

My friend and I sat together to estimate a value of sine

Can u please tell me the name of this method.

Is there any easier estimations than this.

Hi everyone 👋

I've always enjoyed number-sequence puzzles, the "2, 6, 12, 20… what's next?" kind you see in aptitude tests. So I built a free Android app with two parts:

🔢 Solver: Type in any number series and it identifies the pattern. It recognizes 180+ pattern types, including arithmetic, geometric, Fibonacci, polynomial, primes and mixed sequences. It predicts the next terms and explains each step, so you learn why rather than just getting the answer.

🎯 Quizzes: Number-pattern challenges at Easy, Medium and Hard levels, with hints and instant feedback on every answer.

Play store : https://play.google.com/store/apps/details?id=com.patternguru.solver&hl=en_US

Appreciate your thoughts.

The Setup:

Imagine a solid cylinder made of an infinitely soft material with length L and circumference C (radius R = C / 2π).

You place pins at N arbitrary points on the surface of the cylinder and wind a single taut thread around them, starting at an arbitrary point A = (x1, y1) and ending at another arbitrary point B = (x2, y2) anywhere on the cylinder. (Pure geometry setup—ignore gravity, friction, or mass; pulling the two ends of the wire continuously shrinks the surface curve linearly toward the straight interior chord AB.)

Part 1: The Wire Length

• Case 1 (Uniform Winding): Suppose the thread winds uniformly n times around the circumference between A and B (where n does not have to be an integer). What is the total wire length in terms of L, C, n, and the endpoint coordinates (x1, y1) and (x2, y2)?

• Case 2 (General Pinned Path): If the wire connects a sequence of pins P0 = A, P1, ..., Pk = B at coordinates (xi, yi) on the unrolled L × C rectangle, how do you express the segment lengths accounting for paths that wrap across the seam y = 0 = C?

Part 2: The Area of the Cut

Pulling both ends of the wire outward collapses the surface curve linearly through the soft cylinder until it becomes the straight interior line segment connecting A and B.

• How do you express the surface area of the internal slice swept out between the initial curve and the final chord AB in terms of L, C, and the arbitrary endpoints A and B?

Part 3: The Twist

When you pull the wire taut between two distinct endpoints A and B, you might expect it to slice the soft cylinder into two separate pieces—but it never does.

1. Why does the cylinder always remain connected in one piece regardless of L, C, A, B, or how many times the wire wraps around?

2. What special geometric or topological condition on the curve (and endpoints A, B) allows the wire to slice the cylinder completely into two disconnected pieces? (Bonus: How does this cutting condition translate into graph theory if we treat the pins, endpoints, and wire crossings as vertices on the cylinder?)

Hi everyone, I have a cyclic/Latin-square-style counterbalanced repeated-measures design (as shown in image above), and I’m trying to work out the right way to use a Generalized Linear Mixed Effects model to analyse the data for it as I’m new to GLMMs. My aim is to compare how different cell-culture compositions affect the firing rates (expected to be of lognormal or tweedie distribution) of neural cultures while they’re being “trained” to perform tasks.

The study uses 100 neural cultures split evenly into two experiments of 50. Experiment 1 uses Culture Type A and Experiment 2 uses Culture Type B (so culture type is nested within experiment, not crossed). Each experiment’s 50 cultures are prepared in two independent batches of 25, and within every batch the cultures are made up with one of three cell-culture compositions (A, B, or C) - giving four batches and all three compositions in both experiments.

Each culture is then trained under 9 environmental conditions in a counterbalanced order, performing 2 tasks per environmental condition at 600 trials per task.

The GLMM model I currently planning to use is:

Firing rate \\\~ Culture Type + Batch + Cell Culture Composition + Environmental Condition + Task + (1 | Culture ID)

However, I feel that culture ID is insufficient as a random effect and it should include the other variables too, although I am unsure of what combination to use, especially since there’s so many of them. I am also worried about having to add dispersion models in the case of over/under-dispersion as well, since my response variable is not normally distributed. Hence, I would like to seek comments and advice on my current model from those who are experienced in working with GLMMs. Thank you very much in advance for the help.

I believe this should be a division question, however the teacher wrote in that it should be a multiplication question. Was my daughter correct or is the teacher correct?

Do i have enough info on measurements to get the missing measurements on my roof?

If so how would I calculate the 2 measurements of the smaller peak?

I need some advice on what to do in terms of my self-study.
I've been self-studying mathematics for two or three years now, starting with calculus from Paul's Online Math Notes, and working my way up to Tao's Analysis I&II 4th ed, and Gallian's Contemporary Abstract Algebra 11th ed, using Pinter's A Book of Abstract Algebra 2nd ed as supplement and for exercises. However, I'm honestly just struggling and I have no idea what to do at this point.

1. In terms of Analysis, I've been studying around chapter 3-4 of Tao's analysis, but I feel completely lost every section and I can't do the exercises either, even with the hints. Should I go back to Analysis I and restart?

2. In terms of Algebra, I'm nearing a point where I should be switching to Dummit & Foote, but should I be versed in the language of linear algebra first? I haven't taken an introductory course, but I have learnt abstract linear algbra from Friedberg, Insel and Spence's Linear Algebra 5th ed, but I can barely remember anything from that.

3. In terms of exercises, I used to not do them, and even though I have been using them for a bit now, I still don't know how to "choose" them. Often I see people say you shouldn't do all the exercises, but do enough to understand. However, what counts as enough?

I've only ever interacted with topology for fun, and minimally so. My knowledge kinda ends at the whole hole thing and simple homeomorphisms; bottle ≅ sock, straw ≅ skirt, a button ≅ pants; stuff you can say at a party and at least one person will find it interesting.

Anyways, I work at a campus lab and was giving my work friend the hole talk, and they asked me about poking holes into a hollow box. I thought abt it and came to the conclusion that it still wouldn't have a hole. But I realized I never thought about hollow 3D structures (manifolds?). I still feel like it wouldn't have any holes, but i know they can't be homeomorphic. What's the key property/measurement that you would use to explain the difference? I'm sure this question is the trivial-est of trivial, but I've always had a hard time grasping the vocabulary used in topology. I wanna say that the hollow box ≅ 2-sphere and then "poking a hole" in it makes it a,,, a 2-manifold with a 1-manifold boundary?

It might just be better to ask for beginner textbook recommendations.

in this question im using the condition mod (r1-r2) < C1C2 < r1+r2 but ans is given with less than equal to cases intead of just <,> . it says common points wich is plural so why are we putting the equality part.

Subtraction gave us negative numbers
Division gave us fractions
Sqrt gave us complex number
Have tetration(or higher) brought us anything?

Like x^^i ?

and is there any practical usage of such, other than higher order arithmetics?

I acknowledge this seems awfully random and specific, but my son just watched a video from 2024 about what was then the fullest calculation of pi, and the final digit at that time was a 2. I asked him what he thought the following digit would be and now I’m trying to find out the answer for him.

When I saw this in my midterm test last week, I was really confused about how to solve it. I used quadratic equation (which my class did not learn about it yet) to solve it on my scrap paper and my answer was 1250. I'm really curious about how do you solve it without quadratic equations because I genuinely can't think of any other solutions that qualifies the requirements

Hello everyone, hope you are doing well. Since we have a huge spectrum of people here including high schoolers to professional mathematician, I want to ask you what’s a math theorem(s) that you like the most. Be it any field but the statement, brief explanation of the statement and why you like it would be appreciated.

I personally have a lot of favourite theorems but I’d like to quote this one.
Theorem: Every oriented 3-manifold is parallelizable.
Which means the tangent bundle TM of M\^3 is always trivial.

Do share yours, looking forward to reading yours.

Say you want to compress your standard and clunky WGS84 EPSG:4326 location on Planet Earth (e.g. 51°30'45.2"N 0°13'08.6"W) to a single base n integer with as much precision as given number of digits allow. A trivial first step would be to express your latitude and longitude in turns (tr/pla) from the meridian/equator, e.g. 90°W lat would become 0.75 in base10 or 0.C in base16, do the same for lon and extract the needed number of digits.

This approach has two caveats: at first, it is one bit redundant. Both numbers would represent position on a 360° circle, and the second one only needs to hold 180°. Second, needed number of digits for a given precision should be counted for the equator, the worse-case scenario, and for places closer to poles where the circle of latitude is smaller the last digits would be wasted on unnecessary precision.

Is there a formula to convert decimal coordinates to a perfectly efficient representation for an arbitrary base?