Solve Equation: Get f(t) from Af(t)+Bf(t)^C=Dsin(ωt)

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Homework Help Overview

The discussion revolves around solving the equation ##Af(t)+Bf(t)^C=Dsin(\omega t)## for ##f(t)##, where the coefficients A, B, C, and D are real numbers. Participants are exploring the implications of the equation's structure and the conditions under which solutions may be found.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Some participants express uncertainty about how to approach the problem, while others discuss the possibility of solving the equation under specific conditions for C. There is mention of needing numerical methods or approximations for certain values of C, and the complexity of closed-form solutions is highlighted.

Discussion Status

Participants are actively engaging with the problem, questioning the existence of general closed forms for the equation based on the value of C. Some suggest numerical methods as a potential approach, while others reference the Abel-Ruffini theorem in relation to the limitations of closed-form solutions.

Contextual Notes

There is an acknowledgment that the coefficients A, B, C, and D are all real numbers, which influences the discussion on the solvability of the equation.

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Homework Statement


Get ##f(t)## from ##Af(t)+Bf(t)^C=Dsin(\omega t)##.


Homework Equations





The Attempt at a Solution



Sorry guys, but I have no idea what to do :/
 
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skrat said:

Homework Statement


Get ##f(t)## from ##Af(t)+Bf(t)^C=Dsin(\omega t)##.


Homework Equations





The Attempt at a Solution



Sorry guys, but I have no idea what to do :/

So you want to solve the equation ##ax + b x^c = r## to get ##x##. You can do it if ##c \in \{ 1/4, 1/3, 1/2, 1, 2, 3, 4 \}## because there are formulas for solving linear, quadratic, cubic and quartic equations. However, in other cases you need to use approximations or numerical methods. Even for ##c \in \{1/4,1/3,3,4 \}## the formulas are lengthy and nasty, but at least they exist.
 
The only thing I know about A,B,C and D (and sorry for not mentioning that in the first post) is that they are all ##\mathbb{R}##
 
skrat said:
The only thing I know about A,B,C and D (and sorry for not mentioning that in the first post) is that they are all ##\mathbb{R}##

So a general closed form does not exist.
 
Ok, Ray Vikcson, is there a proof for that?

I will try to do it numerically, thanks to both!
 
skrat said:
Ok, Ray Vikcson, is there a proof for that?

I will try to do it numerically, thanks to both!

If you meant a proof for that such expression with ##C\in\{5,6,7,\ldots\}## don't have a general closed form then look at the Abel-Ruffini theorem.
 

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