Two coupled, second order differential equations

In summary, the paper discusses the derivation of normal modes of oscillation for a liquid sphere, where two second order differential equations in two variables are combined into one fourth order equation in one variable. This is achieved by dividing the equations by certain constants and taking derivatives to eliminate certain terms. The paper also asks for a general method for transforming two coupled second order equations into one fourth order equation.
  • #1
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While studying the derivation of the normal modes of oscillation of a liquid sphere in the paper "Nonradial oscillations of stars" by Pekeris (1938), which can be found here, on page 193 and 194 two coupled second order differential equations in two variables are merged into one fourth order differential equation in one variable. I really can't get my head around the way you eliminate one of the two variables.

The two second order equations are:

[itex]c^2\ddot{X}+\dot{X}\left[\dot{c}^2-\left(\gamma-1\right)g+\frac{2c^2}{r}\right]+X\left\{\sigma^2+\left(2-\gamma\right)\left(\dot{g}+\frac{2g}{r}\right)-n\left(n+1\right)\frac{c^2}{r^2}\right\}=g\ddot{w}+\dot{w}\left(2\dot{g}+\frac{2g}{r}\right)+\left[2-n\left(n+1\right)\right]\frac{wg}{r^2}[/itex]

[itex]\dot{X}r^2+X\left[2r+\left(g-\gamma g-\dot{c}^2\right)\left(n+1\right)\frac{n}{\sigma^2}\right]=r^2\ddot{w}+4r\dot{w}+w\left[2-n\left(n+1\right)\right][/itex]

In these equations, all variables depend on [itex]r[/itex] except [itex]\gamma[/itex], [itex]n[/itex] and [itex]\sigma[/itex], which are constants.

Apparently, according to the paper, this can be written as a single, fourth order differential equation in [itex]X[/itex]:

[itex]\ddot{G}+\dot{G}\left(\frac{6}{r}-2\frac{\dot{A}}{A}\right)+G\left\{-\frac{\ddot{A}}{A}+\left(\frac{6-n-n^2}{r^2}\right)-\frac{6 \dot{A}}{Ar}+\frac{2\ddot{A}^2}{A^2}\right\}-AH=0[/itex]

Where

[itex]A=2\left(\frac{\dot{g}}{g}-\frac{1}{r}\right)[/itex]

[itex]gG=c^2\ddot{X}+\dot{X}\left(\dot{c}^2-\gamma g+\frac{2c^2}{r}\right)+X\left[\sigma^2+\left(2-\gamma\right)\dot{g}+\left(1-\gamma\right)\frac{2}{r}-n\left(n+1\right)\frac{c^2}{gr^2}+\frac{n}{\sigma^2r^2}\left(n+1\right)\left(\dot{c}^2-g+\gamma g\right)\right][/itex]

[itex]H=\ddot{X}+\dot{X}\left[\frac{4}{r}-\frac{n}{\sigma^2 r^2}\left(n+1\right)\left(\dot{c}^2-g+\gamma g\right)\right]+X\left[\frac{2}{r^2}-\frac{n}{\sigma^2r^2}\left(n+1\right)\left(\ddot{c}^2-\dot{g}+\gamma\dot{g}\right)\right][/itex]

Does someone know a general way to transform two second order coupled differential equations into one fourth order equation? Thanks for any hints or help!
 
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  • #2
Divide the first equation by g and the second by [itex]r^2[/itex] and subtract both equations to get an equation (eq. 3.) with the [itex]\ddot{w}[/itex] term eliminated. Take the derivative of this equation (3.) to eliminate w: [itex]w \rightarrow \dot{w}[/itex] and [itex]\dot{w} \rightarrow \ddot{w}[/itex]. Do the same again to eliminate the new [itex]\ddot{w}[/itex] term in eq. 3. Take the derivative of eq. 3. so [itex]\dot{w} \rightarrow \ddot{w}[/itex]. Do the previous once more to get an equation for [itex]X^{(IV)}[/itex] with all the w-terms eliminated.
 

What are two coupled, second order differential equations?

Two coupled, second order differential equations are a set of two equations that involve second derivatives of two dependent variables. These equations are linked or "coupled" by one or more terms containing both dependent variables.

What is the purpose of solving two coupled, second order differential equations?

The purpose of solving two coupled, second order differential equations is to find the relationship between the two dependent variables and their rates of change over time. This is useful in many fields of science, such as physics, engineering, and biology, as it allows for predicting and understanding the behavior of complex systems.

How are two coupled, second order differential equations solved?

Two coupled, second order differential equations are typically solved using numerical methods, such as the Euler method or the Runge-Kutta method. These methods involve breaking down the equations into smaller, simpler steps and approximating the solution at each step.

What are some real-world applications of two coupled, second order differential equations?

Two coupled, second order differential equations have many real-world applications, including modeling the motion of objects under the influence of multiple forces, predicting the behavior of electrical circuits, and studying the dynamics of chemical reactions.

What are some challenges in solving two coupled, second order differential equations?

One of the main challenges in solving two coupled, second order differential equations is finding the initial conditions, or the values of the dependent variables and their derivatives at a specific starting point. Additionally, the complexity of the equations and the need for numerical methods can make the solutions computationally intensive and time-consuming.

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