JackDaniel87
New member
- Joined
- Dec 9, 2021
- Messages
- 6
Hey guys,
Nice to be on here.
I have been banging my brain for the last two weeks trying to come up with an algebraic solution to the following question - to no avail.
Any input would be MUCH appreciated!
The problem is somewhat long but can be summarized as follows:
Begin with the following equation as a function of x. There are two parameters, a and b, that could take on arithmetic values but I am more interested in a general solution:
[math]y(x)=\frac{b\left(1-x\right)}{b\left(1-x\right)+\left(1-a\right)x}[/math]
The curvature K of the above polynomial ought to be given by the the following differential equation which uses the first and second order derivatives of y(x), as follows:
[math]?=\frac{|?″|}{{(1+?′^2)}^\frac{3}{2}}[/math]
Now, I am actually interested in the maximum curvature k - which is why we need to differentiate k with respect to x and find its roots:
[math]?′=\frac{d}{dx}k[/math]
Hence, I am interested in finding the roots of k' as a function of a and b, particularly for values of x between 0 and 1. I know a solution exists because graphically it is evident, as seen here, where the purple line (k') crosses the x-axis:
However, obtaining an algebraic solution as a function of a and b has been a challenge - hence my reaching out!
Any input you might have would be GREATLY appreciated!
Thank you in advance for any help you may offer!
-J
Nice to be on here.
I have been banging my brain for the last two weeks trying to come up with an algebraic solution to the following question - to no avail.
Any input would be MUCH appreciated!
The problem is somewhat long but can be summarized as follows:
Begin with the following equation as a function of x. There are two parameters, a and b, that could take on arithmetic values but I am more interested in a general solution:
[math]y(x)=\frac{b\left(1-x\right)}{b\left(1-x\right)+\left(1-a\right)x}[/math]
The curvature K of the above polynomial ought to be given by the the following differential equation which uses the first and second order derivatives of y(x), as follows:
[math]?=\frac{|?″|}{{(1+?′^2)}^\frac{3}{2}}[/math]
Now, I am actually interested in the maximum curvature k - which is why we need to differentiate k with respect to x and find its roots:
[math]?′=\frac{d}{dx}k[/math]
Hence, I am interested in finding the roots of k' as a function of a and b, particularly for values of x between 0 and 1. I know a solution exists because graphically it is evident, as seen here, where the purple line (k') crosses the x-axis:
However, obtaining an algebraic solution as a function of a and b has been a challenge - hence my reaching out!
Any input you might have would be GREATLY appreciated!
Thank you in advance for any help you may offer!
-J
I looked at the general expression for [imath]k^\prime[/imath] and came to the conclusion that if [imath]k^\prime=0[/imath] then [imath]\left(1+{y^\prime}^2\right)y^{\prime\prime\prime}-3{y^{\prime\prime}}^2=0[/imath]. I am not too confident in the last equation, so feel free to double check. Even if I haven't made any mistakes in my derivation I am not sure it is a helpful result: when you plugin your expression for [imath]y[/imath] you will most likely end up with high order polynomials where closed-form solutions might not exist.