You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an alternative browser.
You should upgrade or use an alternative browser.
Question about a Differential equation
- Thread starter Navid555
- Start date
D
Deleted member 4993
Guest
This is a Nonlinear second order ordinary differential equation. Have you studied Duffing's equation in vibration? If not use Google and look it up.Hello. This a differential equation and i have some question about it.
1.is this ode type or pde?
2.how i should solve it? consider zeta defined with x and t parameter.
View attachment 30752
Please show us what you have tried and exactly where you are stuck.
Please follow the rules of posting in this forum, as enunciated at:
Please share your work/thoughts about this problem.
Last edited by a moderator:
Thank you. Is it for ode equation or pde?It can be solved in terms of the Jacobi elliptic function.
Thank you. How can i write this equation as derivative to x and t separately instead of zetaIt can be solved in terms of the Jacobi elliptic function.
We have \(\displaystyle \theta(\zeta(x,t))\).
The first derivative will be
\(\displaystyle \frac{d\theta}{d\zeta} = \frac{\partial \theta}{\partial x} \frac{dx}{d\zeta} + \frac{\partial \theta}{\partial t} \frac{dt}{d\zeta} = \frac{\partial \theta}{\partial x} - \frac{\partial \theta}{\partial t} \frac{1}{a}\)
The second derivative will be
\(\displaystyle \frac{d^2\theta}{d\zeta^2} = \frac{\partial}{\partial x}\left(\frac{\partial \theta}{\partial x} - \frac{\partial \theta}{\partial t} \frac{1}{a}\right)\frac{dx}{d\zeta} + \frac{\partial}{\partial t}\left(\frac{\partial \theta}{\partial x} - \frac{\partial \theta}{\partial t} \frac{1}{a}\right)\frac{dt}{d\zeta}\)
\(\displaystyle = \frac{\partial^2 \theta}{\partial x^2} - \frac{\partial^2 \theta}{\partial x\partial t} \frac{1}{a} - \frac{\partial^2 \theta}{\partial t\partial x}\frac{1}{a} + \frac{\partial^2 \theta}{\partial t^2} \frac{1}{a^2}\)
\(\displaystyle = \frac{\partial^2 \theta}{\partial x^2} - \frac{\partial^2 \theta}{\partial x\partial t}\frac{2}{a} + \frac{\partial^2 \theta}{\partial t^2} \frac{1}{a^2}\)
Substituting this in the original DE, you will get
\(\displaystyle \frac{\partial^2 \theta}{\partial x^2} - \frac{\partial^2 \theta}{\partial x\partial t}\frac{2}{a} + \frac{\partial^2 \theta}{\partial t^2} \frac{1}{a^2} = c_1\theta + c_2\theta^3\)
The first derivative will be
\(\displaystyle \frac{d\theta}{d\zeta} = \frac{\partial \theta}{\partial x} \frac{dx}{d\zeta} + \frac{\partial \theta}{\partial t} \frac{dt}{d\zeta} = \frac{\partial \theta}{\partial x} - \frac{\partial \theta}{\partial t} \frac{1}{a}\)
The second derivative will be
\(\displaystyle \frac{d^2\theta}{d\zeta^2} = \frac{\partial}{\partial x}\left(\frac{\partial \theta}{\partial x} - \frac{\partial \theta}{\partial t} \frac{1}{a}\right)\frac{dx}{d\zeta} + \frac{\partial}{\partial t}\left(\frac{\partial \theta}{\partial x} - \frac{\partial \theta}{\partial t} \frac{1}{a}\right)\frac{dt}{d\zeta}\)
\(\displaystyle = \frac{\partial^2 \theta}{\partial x^2} - \frac{\partial^2 \theta}{\partial x\partial t} \frac{1}{a} - \frac{\partial^2 \theta}{\partial t\partial x}\frac{1}{a} + \frac{\partial^2 \theta}{\partial t^2} \frac{1}{a^2}\)
\(\displaystyle = \frac{\partial^2 \theta}{\partial x^2} - \frac{\partial^2 \theta}{\partial x\partial t}\frac{2}{a} + \frac{\partial^2 \theta}{\partial t^2} \frac{1}{a^2}\)
Substituting this in the original DE, you will get
\(\displaystyle \frac{\partial^2 \theta}{\partial x^2} - \frac{\partial^2 \theta}{\partial x\partial t}\frac{2}{a} + \frac{\partial^2 \theta}{\partial t^2} \frac{1}{a^2} = c_1\theta + c_2\theta^3\)