Trajectories of a linear system first order diff. equations

In summary, a "trajectory graph" shows multiple solutions to a system of differential equations with different initial conditions, while a "slope field" shows the slope of a single solution at each point.
  • #1
Nikitin
735
27
In linear algebra, you can have systems of differential equations represented by matrices.

What does a "trajectory graph" of such a system show, exactly? And how can you draw one?

What's the difference between such a trajectory-graph and an ordinary slope-field for a single linear differential equation?

Thanks! :)
 
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  • #2
The "slope field" is a graph showing short straight lines at each (x, y) point (well, in reality as many as possible without one covering another!), showing the slope of a solution to the differential equation. A "trajectory graph" is a graph showing an number of actuals "trajectories" (solutions) of the differential equation, with different initial conditions as possible.

If you have a "slope field" for a differential equation, each of the trajectories must be parallel to the slope field lines at each point.
 

1. What is a linear system of first-order differential equations?

A linear system of first-order differential equations is a set of equations that describe the relationship between a set of variables and their rates of change. Each equation is first-order, meaning it involves only first derivatives, and the system is linear, meaning the variables are directly proportional to each other.

2. How are trajectories of a linear system of first-order differential equations determined?

The trajectories of a linear system are determined by solving the differential equations. This involves finding a set of functions that satisfy all of the equations simultaneously. These functions represent the paths that the system's variables will follow over time.

3. What factors affect the trajectories of a linear system?

The trajectories of a linear system can be affected by a variety of factors, including the initial conditions of the system, the values of the system's parameters, and any external forces or inputs acting on the system.

4. Can the trajectories of a linear system be predicted?

Yes, the trajectories of a linear system can be predicted by solving the differential equations and using the known initial conditions and parameters. However, the accuracy of the predictions may depend on the complexity of the system and the accuracy of the initial conditions and parameters.

5. How are linear systems of first-order differential equations used in scientific research?

Linear systems of first-order differential equations are used in various fields of science to model and analyze the behavior of complex systems. They can be used to study the dynamics of physical systems, population growth, chemical reactions, and many other phenomena. They are also commonly used in engineering and economics to make predictions and optimize systems.

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