Maximus2023
New member
- Joined
- Apr 15, 2023
- Messages
- 2
Hello! There is Duffing’s equation in my problem (first DE in my problem)
\(\displaystyle (df/dx)^2 - af^2 + bf^4 / 2 = c, \)
where a<0, b<0 and c>0 are constants, f =f(x), x is 1-D real coordinate.
Its solution is an elliptic sine
\(\displaystyle f(x) = A \text{sn}(Bx, k^2),\)
where A, B and k (k is a modulus) are constants of a, b and c.
But in my problem f=f(x,t), where t is time, therefore c=c(t) and therefore
\(\displaystyle (\partial f/\partial x)^2 - af^2 + bf^4 / 2 = c(t), \)
and
\(\displaystyle f(x,t) = A(c(t)) \text{sn}(B(c(t))x, k^2(c(t))). \)
There is a second DE in my problem, which contains \(\displaystyle \partial f(x,t) / \partial t \). It means one have to calculate
\(\displaystyle \partial \text{sn}(Bx, k^2) / \partial k^2, \)
but I don’t know how to do it (I did’t find anything about it) and am looking here for help with it.
Thank you
\(\displaystyle (df/dx)^2 - af^2 + bf^4 / 2 = c, \)
where a<0, b<0 and c>0 are constants, f =f(x), x is 1-D real coordinate.
Its solution is an elliptic sine
\(\displaystyle f(x) = A \text{sn}(Bx, k^2),\)
where A, B and k (k is a modulus) are constants of a, b and c.
But in my problem f=f(x,t), where t is time, therefore c=c(t) and therefore
\(\displaystyle (\partial f/\partial x)^2 - af^2 + bf^4 / 2 = c(t), \)
and
\(\displaystyle f(x,t) = A(c(t)) \text{sn}(B(c(t))x, k^2(c(t))). \)
There is a second DE in my problem, which contains \(\displaystyle \partial f(x,t) / \partial t \). It means one have to calculate
\(\displaystyle \partial \text{sn}(Bx, k^2) / \partial k^2, \)
but I don’t know how to do it (I did’t find anything about it) and am looking here for help with it.
Thank you