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qetuol
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my question is: what is the general solution of this system of coupled diff. equations:
f ''i = Cijfj
C is a matrix, fj(z) are functions dependent of z.
f ''i = Cijfj
C is a matrix, fj(z) are functions dependent of z.
elibj123 said:It will be almost the same as a first-order equation.
Guess
[tex]\vec{f}(t)=\vec{f}_{s}e^{st}[/tex]
Plug in and get:
[tex]s^{2}\vec{f}_{s}e^{st}=C \vec{f}_{s}e^{st}[/tex]
[tex](C-s^{2}I)\vec{f}=0[/tex]
So here s^2 are the eigenvalues of C. And each eigenvalue has two corresponding solutions with s and -s. I haven't seen treatment of different cases (complex s, multiplicity and such) but I guess it'll be the same as with the first-order analysis.
A system of second order linear homogenous differential coupled equations refers to a set of two or more equations that involve derivatives of the same dependent variable, with no constant term, and where the coefficients of the derivatives are constant. This type of system is called "homogenous" because all the equations have the same dependent variable and "coupled" because the equations are connected through the dependent variable.
The main difference is that these equations involve more than one dependent variable and they are connected through derivatives. This means that the solutions to these equations will be a combination of functions for each dependent variable, rather than a single function for a single dependent variable. It also means that the solutions can be more complex and require more advanced mathematical techniques to solve.
These types of equations are commonly used in physics, engineering, and other scientific fields to model complex systems. They can be used to study the motion of particles, the behavior of circuits, and the dynamics of chemical reactions, among other things.
The general method involves finding the characteristic equation of the system, which is a polynomial equation that relates the coefficients of the derivatives to the dependent variables. This equation can then be solved using various techniques, such as factoring, substitution, or using the quadratic formula. The solutions to the characteristic equation will give the general form of the solutions to the system of equations.
One special case is when the coefficients of the derivatives are constant and the equations are uncoupled, meaning they do not involve derivatives of the same dependent variable. In this case, the solutions can be found by solving each equation separately. Another technique is using Laplace transforms, which can simplify the solutions to these equations and make them easier to solve in certain cases.