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

[skrat]

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 :/

 

[Ray Vickson]

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.

 

[skrat]

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 $\mathbb{R}$

 

[HakimPhilo]

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.

 

[skrat]

Ok, Ray Vikcson, is there a proof for that?

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

 

[HakimPhilo]

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.