System of Second Order, Nonhomogeneous Differential Equations

In summary, an engineering student is seeking help with solving a system of second order, nonhomogeneous equations involving masses, damping coefficients, and spring constants. They are also required to write a MATLAB program to solve the problem and implement a numerical method. The problem involves 3 masses hanging by springs and dashpots, and the student needs to write the differential equations and solve them using a self-written m-file. The student also needs to define a new vector and rewrite the equations before solving the ODE set.
  • #1
mccormas
2
0
Hello. I am an engineering student and am having trouble trying to figure out how to solve this system of second order, nonhomogeneous equations. I know how to solve a single second order, nonhomo. equation and how to solve a system of first orders, but not this one. Any help would be greatly appreciated. I have attached a scanned image of the equation because I couldn't format it right on here and it won't let me just post a URL because I'm new on here.


That is a g tucked in there on the right side of the equation. Thanks again.
 

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  • #2
keeping in mind that you are an engineering student, I would speculate and assume that g is gravity, the masses are hanged from somewhere. and in general, you are required the solve the x(t) for each mass.
 
  • #3
Exactly. I probably should have specified that earlier. The m's are masses, c's are damping coefficients and k's are spring constants. All are positive constants that will be specified by the user in the MATLAB program that I have to write. First, though, I need to figure out how to solve the problem by hand. Then I have to choose a numerical method we've learned and implement it in an M-file on MATLAB.

The problem has 3 masses hanging by springs and dashpots. We have to write the differential equations for the system and then solve it using a self-written m-file.
 
  • #4
Then, define a new vector,

[tex]
\hat x = \left( {\begin{array}{*{20}c}
x \\
{\dot x} \\
\end{array}} \right)

[/tex]

From the fact
[tex]
\frac{d}{dt}\hat x = \left( {\begin{array}{*{20}c}
\dot{x} \\
{\ddot x} \\
\end{array}} \right)

[/tex]

Rewrite the equations again. and then solve a ODE set.
 

1. What is a system of second order, nonhomogeneous differential equations?

A system of second order, nonhomogeneous differential equations is a set of mathematical equations that describe the behavior of a dynamic system over time. These equations involve second order derivatives and do not have a constant solution, meaning they are not easily solvable by hand.

2. How is a nonhomogeneous differential equation different from a homogeneous one?

A nonhomogeneous differential equation contains a forcing function, which is a term that is not dependent on the variable being solved for. This forcing function introduces external influences on the system, while a homogeneous differential equation does not have any external influences.

3. How do you solve a system of second order, nonhomogeneous differential equations?

There are various methods for solving a system of second order, nonhomogeneous differential equations, such as the method of undetermined coefficients, variation of parameters, and Laplace transforms. These methods involve finding a particular solution and a general solution to the equations, and then combining them to get the final solution.

4. What are some real-life applications of systems of second order, nonhomogeneous differential equations?

Systems of second order, nonhomogeneous differential equations are commonly used in fields such as physics, engineering, and economics to model and predict the behavior of complex systems. They can be used to study the motion of objects, the flow of fluids, and the growth of populations, among other things.

5. Can a system of second order, nonhomogeneous differential equations have multiple solutions?

Yes, a system of second order, nonhomogeneous differential equations can have multiple solutions depending on the initial conditions and the specific method used to solve the equations. These solutions can also vary in complexity and may involve trigonometric or exponential functions.

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