How can I graph systems of differential equations with Mathematica?

[b2386]

Hi all,

Right now I am doing some Diff equations homework but am having difficulty with a problem. The problem asks me to draw a direction field for the system $$x' = \left(\begin{array}{cc}1 & 1\\4 & 1\end{array}\right)\left(\begin{array}{cc}x_1\\x_2\end{array}\right)$$

The proper Mathematica function is of the form PlotVectorField[{f(x), f(y)},{x, xmin, xmax}, {y, ymin, ymax}], where f(x) and F(y) are the function in terms of x and y. I understand how to do this for a simple function such as $$y' -2y=4-t$$. In this case, f(x) = 1 and f(y) = 2y-t+4. However, the matrices that I am currenly facing make things for difficult. The resulting graph should have axes of x_1 and x_2 but that's all I know for sure. Can anyone help me please?

 

[mmwave]

Hello,

I can understand your struggles with drawing a direction field for a system with matrices. Here are some steps that can help you with this problem:

1. First, let's understand what a direction field represents. It is a graphical representation of the slope of a solution to a differential equation at a particular point. In this case, we have a system of equations, so we will have multiple slopes at each point.

2. To plot a direction field in Mathematica, we need to use the PlotVectorField function as mentioned in the forum post. This function requires two inputs: the vector field and the domain.

3. The vector field in this case is the system of equations itself, which is x' = Ax, where A is the matrix given in the problem. So, we need to find a way to represent this in terms of x and y.

4. One way to do this is by using the eigenvalues and eigenvectors of the matrix A. The eigenvalues will give us the slopes of the solutions, and the eigenvectors will give us the direction of those slopes.

5. Once we have the eigenvalues and eigenvectors, we can use them to construct the vector field in terms of x and y. This can be done using the matrix multiplication and some algebraic manipulation.

6. Finally, we can use the PlotVectorField function to plot the direction field using the vector field we just constructed and the desired domain.

I hope this helps you with your problem. If you need more guidance, I suggest consulting your textbook or seeking help from your instructor or a tutor. Good luck with your homework!

 

[tacman]

Hi there,

Graphing with Mathematica can definitely be tricky, especially when dealing with matrices. To draw a direction field for your system of differential equations, you can use the function StreamPlot[{f(x), f(y)}, {x, xmin, xmax}, {y, ymin, ymax}]. In this case, f(x) and f(y) would represent the two equations in your system. For example, if your system was x' = 3x + 2y and y' = 2x + y, then f(x) = 3x + 2y and f(y) = 2x + y. The resulting graph will show you the direction of the vector at each point (x,y) in the given range. I hope this helps and good luck with your homework!