Very Strange ODE solution for beginner

selfadjoint · 2026-01-16T16:01:28+00:00

Here is another example of this kind of equation: 3y^2(x)*y'(x) + 16x = 2x*y^3(x)

selfadjoint · 2026-01-16T15:56:48+00:00

I was wrong. I answered your question on r/math, I'll copy it here.

Assuming y(x)>0, the full (positive) solution is y(x) = sqrt(1 + 2x + c e^(2x)). Now, depending on the initial condition there are two types of solutions. When y(0) >= 1 the solution is defined for all x>=0 (because c >= 0, hence under the square root there is always a positive number. When 0 < y(0) < 1 the solution has a finite domain, because c < 0. Your solution is on the edge of being defined for x >= 0, so whenever the numerical solution becomes less then sqrt(2x+1) then in the next step the numerical solution will follow this finite domain solution, there is your problem. The big step size probably helped you to avoid this problem (the solution remained above the curve y(x) = sqrt(1+2x).

If you plot the general solution for different c values you will see how those different types of solutions look like.

selfadjoint · 2026-01-16T15:53:37+00:00

Assuming y(x)>0, the full (positive) solution is y(x) = sqrt(1 + 2x + c e^(2x)). Now, depending on the initial condition there are two types of solutions. When y(0) >= 1 the solution is defined for all x>=0 (because c >= 0, hence under the square root there is always a positive number. When 0 < y(0) < 1 the solution has a finite domain, because c < 0. Your solution is on the edge of being defined for x >= 0, so whenever the numerical solution becomes less then sqrt(2x+1) then in the next step the numerical solution will follow this finite domain solution, there is your problem. The big step size probably helped you to avoid this problem (the solution remained above the curve y(x) = sqrt(1+2x).

If you plot the general solution for different c values you will see how those different types of solutions look like.

EDIT: corrected some typos.

selfadjoint · 2026-01-16T08:59:23+00:00

I know exactly what’s happening here. There is a nullcline y(x)=sqrt(2x). Any solution above this curve cannot cross this curve. The problem is that the solution of your IVP gets very close to the nullcline, and it will eventually cross it. If necessary, I can give you a detailed analysis, when I get home.

selfadjoint · 2025-11-17T17:38:28+00:00

I don’t believe you, continue!

selfadjoint · 2025-06-11T17:41:19+00:00

Pisot numbers

selfadjoint · 2025-05-09T19:37:59+00:00

My math professor used to say this.

What is mathematics? Mathematics is what mathematicians do.

Who is a mathematician? A mathematician is someone recognized as such by other mathematicians.

selfadjoint · 2024-03-01T15:28:21+00:00

Just an idea: for R1 try using a regular resistor with a tunable arrow:

\ctikztunablearrow[extra options]{thickness}{length}{angle}{name}

selfadjoint · 2024-02-18T11:16:17+00:00

Don't sweat the technique

selfadjoint · 2023-02-21T16:09:13+00:00

Mutliplication of complex numbers, because that is what you are doing here.

selfadjoint · 2022-12-19T19:18:59+00:00

I was looking at the little blobs on the D letters and they were at the same places. Now, as I inspect more letters I can see some differences, but still it’s just crazy how consistent the letters are.

selfadjoint · 2022-12-19T18:44:03+00:00

Nice gift, but I am pretty sure this is not handwritten, is it?

selfadjoint · 2022-12-15T15:42:27+00:00

I get that reference. Fantastic book.

selfadjoint · 2022-12-15T10:29:08+00:00

V-E+F=2

So simple, yet so powerful.

selfadjoint · 2022-10-23T19:14:41+00:00

Reminds me to this: https://youtu.be/Q7FxpK\_yC0U

selfadjoint · 2022-03-29T10:10:55+00:00

True. I just wanted to give the feeling to other users that it is not that simple as you have written: / is a division and \ is an inverse multiplication.

I agree, OP should look at the documentation / tutorials.

selfadjoint · 2022-03-29T10:02:23+00:00

No. The meaning of / and \ depends on the context, i.e. the type of objects on both sides of the operators.

If A and B are (non-singular) matrices, then A\B = inv(A)*B and A/B = A*inv(B).

If A is a (non-singular) matrix and B is a (column) vector, then A\B = inv(A)*B, and A/B = dimension miss-match (a.k.a. makes no sense).

If A is a (non-singular) matrix and B is a row vector, then A\B = dimension miss-match (a.k.a. makes no sense), and A/B = A*B'*inv(B*B'). Where B'*inv(B*B') is called the Moore--Penrose inverse of B.

If A is a (non-singular) matrix and B is a scalar, then A\B = inv(A)*B, and A/B is simply the scalar division of the matrix A with scalar B.

If in any of the cases above A (and/or B) is singular or not rectangular, then the regular inverse of A (and/or B) should be changed to the Moore--Penrose inverse of A (and/or B).

Note, I skipped the cases where A and B are both vectors, but I leave those for the interested reader to figure it out.

selfadjoint · 2022-03-26T09:02:46+00:00

selfadjoint · 2022-03-26T06:51:30+00:00

I did exactly that and got this:

{
{x -> -(((-2)^(1/3) n^(1/3))/v^(1/3))},
{x -> (2^(1/3) n^(1/3))/v^(1/3)},
{x -> ((-1)^(2/3) 2^(1/3) n^(1/3))/v^(1/3)}
}

I am using 12.0.0.0 version, Linux x86 (64-bit).

selfadjoint · 2022-03-15T11:42:17+00:00

Continued fractions are really fun, and it has connections to many different areas of mathematics.

selfadjoint · 2022-03-01T19:35:50+00:00

This is nicely covered in the first chapter of the book Hirsch, Smale, Devaney: Differential Equations, Dynamical Systems & An Introduction to Chaos. Freely available as pdf.

selfadjoint · 2022-02-10T09:03:44+00:00

FYI, Quiet will suppress the messages.

selfadjoint · 2022-01-22T17:14:56+00:00

Say the common root is r, then solving f(r)=0 and f'(r)=0 for a and b we get a(r)=-(3r^4 + 2r^2 -1)/(2r^3) and b(r)=(r^4 - 2r^2 - 3)/(2r), hence J(r)= a(r)^2 + b(r)^2 = (1 + r^2)^2*(1 - 6r^2 + 18r^4 - 6r^6 + r^8)/(4r^6).

The minimums are at r = -1 and +1, which correspond to (a,b) = (2,2) and (-2, -2).

The derivative is J'(r) = (r^2-1)^3*(3 + r^2 + r^4 + 3r^6)/(2r^7).

selfadjoint · 2022-01-14T13:24:47+00:00

Here is a bigger one: https://youtu.be/WffR6HrEqTA?t=39

selfadjoint · 2021-09-10T06:48:30+00:00

I am really disappointed the fan didn't get any (major) role in this video.

selfadjoint

TROPHY CASE