/sci/ - Science & Math » Thread #12710464

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Anonymous Tue 16 Feb 01:05:55 2021 No.12710464 View Reply Original Report

Quoted By:

I just solved this difficult differential equation! I feel smart! Solve this using complex analysis f''(x) = f(f(x)

Anonymous

Anonymous Tue 16 Feb 2021 01:07:13 No.12710467 Report

Quoted By:

i don't believe you. show your work. is it a general solution?

Anonymous

Anonymous Tue 16 Feb 2021 01:08:16 No.12710473 Report

Quoted By:

how about you prove earth isn't flat first and then we can start thinking about your schizo math problems

Anonymous

Anonymous Tue 16 Feb 2021 01:10:17 No.12710483 Report

Quoted By: >>12710494

Bro it's easy. Just take the log. Logarithmic derivative related to log of function. Then use Cauchy equations.

Anonymous

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Anonymous Tue 16 Feb 2021 01:12:30 No.12710494 Report

Quoted By:

>>12710483
cauchy equations dont give you the solution

Anonymous

Anonymous Tue 16 Feb 2021 01:17:37 No.12710517 Report

Quoted By: >>12710568

$\displaystyle \begin{align*} f(t) &= \cos(t)+i \sin(t) \qquad (\Rightarrow f(0)=1) \\ \Rightarrow f'(t) &= - \sin(t)+i \cos(t) \\ &= i( \cos(t)+i \sin(t)) \\ &= if(t) \\ \Rightarrow \frac{f'(t)}{f(t)} &= i \\ \Rightarrow \int_{0}^{x} \frac{f'(t)}{f(t)} \, dt &= \int_{0}^{x}i \, dt \\ \Rightarrow \ln(f(x))- \ln(f(0)) &= i(x-0) \\ \Rightarrow \ln(f(x))- \ln(1) &= ix \\ \Rightarrow \ln(f(x)) &= ix \\ \Rightarrow f(x) &= \exp(ix) \\ \Rightarrow \cos(x)+i \sin(x) &= \exp(ix) \\ \end{align*} $

Anonymous

Anonymous Tue 16 Feb 2021 01:38:35 No.12710568 Report

Quoted By: >>12710587

>>12710517
is this what autism looks like

Anonymous

Anonymous Tue 16 Feb 2021 01:42:31 No.12710587 Report

Quoted By: >>12710593

>>12710568
it's a proof of Euler's Identity...

yes

Anonymous

Anonymous Tue 16 Feb 2021 01:43:41 No.12710593 Report

Quoted By: >>12710603 >>12710661 >>12710716

>>12710587
No it isn't. It uses logarithms before proving you can, and it totally ignores the existence of branch cuts. You can't use logarithms to prove Euler's identity.

Anonymous

Anonymous Tue 16 Feb 2021 01:46:16 No.12710603 Report

Quoted By: >>12710674

>>12710593
damn, can you solve the OP'S problem though?

Anonymous

Anonymous Tue 16 Feb 2021 02:00:52 No.12710661 Report

Quoted By: >>12710692

>>12710593
>f'(x) = f(f(x) is much easier to solve, can you solve it?

Anonymous

Anonymous Tue 16 Feb 2021 02:03:12 No.12710674 Report

Quoted By:

>>12710603
Method of Frobenius is key.

Anonymous

Anonymous Tue 16 Feb 2021 02:08:11 No.12710692 Report

Quoted By: >>12710762

>>12710661
Your problem is formally equivalent to solving the generalized ODE

$\frac{d^{(n)}y}{dx} = \sum_{i=0}^\infty a_i [y(x)]^i$

with either n = 1 or n = 2., where

$y(x) = \sum_{i=0}^\infty a_i x^i$

It can be done algorithmically. It probably comes out to a clean answer at the end.

Anonymous

Anonymous Tue 16 Feb 2021 02:12:55 No.12710716 Report

Quoted By:

>>12710593
>uses logarithms before proving you can
reinventing the wheel

Anonymous

Anonymous Tue 16 Feb 2021 02:27:08 No.12710762 Report

Quoted By:

>>12710692
f(x) = (e^((sqrt(3)/6) *pi) * (sqrt(3)/2 + i/2)) * x^(1/2 + i * sqrt(3) / 2)

^^^^ I'll let you guys try to show my work lol

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