To follow up on this, I had it solve a nasty ODE problem that I saw in the recent Mathematica 15 release post:

    Solve the following first-order ODE for f(x):

    ((-1 - 2*x)*f(x)*tan(1 + x - exp(-61 - 2*x)*f(x)/x)
    + exp(61 + 2*x)*x*(1 - x*tan(1 + x - exp(-61 - 2*x)*f(x)/x))
    + x*tan(1 + x - exp(-61 - 2*x)*f(x)/x)*f'(x)) = 0

    Find the general solution f(x).

And surprisingly it found a valid solution! Extra impressive because it runs 25 tok/s on my measly RTX 2070 super.

    f(x) = x*exp(61 + 2*x)*(1 + x - arccos(C/x))

    C is an arbitrary constant.

Apparently Mathematica 14.3 couldn't solve this ODE.

Qwen 3.5 9B Q4_K_M solved this using 10K tokens in 5 mins on a RX 7600.

The answer is exactly what you have posted. I am impressed by Qwen!

How do you know it’s a valid solution? Are you able to verify it yourself?

This is a math problem with a math solution. You verify it with math

Interesting!

I just tried the quantized Q4_K_M from [1] in my RTX 2070 Super, it ran at 110 tok/s with 1800 tok/s prefill, and found the same solution to your prompt. It generated valid LaTeX for the answer but its reasoning trace uses mostly compact ASCII math notation. Took 3min 22s to answer, spending 22k tokens almost all on thinking.

[1] https://huggingface.co/prithivMLmods/VibeThinker-3B-GGUF

How do we know the solution isn't in the weights though?

If it can code well then once you put it in a loop with an interpreter it can do anything.