Trace vector directions, flow, and solution paths. Adjust bounds, steps, and start points with ease. Save clean outputs, tables, and plots for deeper analysis.
This example uses dx/dt = y and dy/dt = -x with start point (1, 0).
| t | x | y | Meaning |
|---|---|---|---|
| 0.00 | 1.0000 | 0.0000 | Starting point on the phase curve |
| 1.57 | 0.0008 | -1.0000 | Quarter turn of the orbit |
| 3.14 | -1.0000 | -0.0016 | Half turn around the center |
| 4.71 | -0.0024 | 1.0000 | Three quarter turn |
| 6.28 | 1.0000 | 0.0032 | Approximate return to the start |
A vector field in the plane is written as:
dx/dt = f(x, y, t)
dy/dt = g(x, y, t)
The local direction at any point comes from the vector (f, g).
The local slope of the trajectory is:
dy/dx = (dy/dt) / (dx/dt), when dx/dt ≠ 0.
This calculator uses the fourth order Runge Kutta method for numerical integration.
For each step h:
k1 = F(x, y, t)
k2 = F(x + h·k1/2, y + h·k1/2, t + h/2)
k3 = F(x + h·k2/2, y + h·k2/2, t + h/2)
k4 = F(x + h·k3, y + h·k3, t + h)
The next point is the weighted average of these four direction estimates. This improves stability and plot smoothness.
A vector field trajectory plotter helps you study how a system moves over time. Each point in the plane has a direction. That direction comes from the differential equations. The plotted curve follows those directions step by step. This makes the tool useful for phase plane analysis, flow visualization, and dynamical system exploration. It can also help students see how initial conditions change the shape of a solution curve.
This calculator is designed for two dimensional first order systems. You enter dx/dt and dy/dt, then choose a starting point. The tool computes the trajectory across a time interval. It also draws the surrounding field so the motion becomes easier to read. This is helpful in maths courses, numerical methods work, control problems, and nonlinear system studies. It also supports time dependent expressions through the t variable.
The result section reports more than a line on a graph. It shows the final point, starting slope, speed, displacement, and arc length. These extra values support quick interpretation. A smaller step size usually gives a smoother path. A larger plot window helps you inspect global behaviour. A denser field shows more local direction detail. The CSV export is useful for reports, checking values, or using the data in another workflow.
Many learners understand formulas but struggle with geometric meaning. A trajectory plot turns symbolic equations into visible motion. You can identify spirals, closed loops, saddle like movement, or paths that approach equilibrium. You can also compare systems by changing coefficients and observing the new curve. Because the calculator keeps the layout simple, the focus stays on the mathematics. This makes it a strong tool for teaching, revision, and quick exploratory analysis.
It is the path traced by a solution of a differential system. The curve follows the local direction vectors defined by dx/dt and dy/dt at each point.
They describe how x and y change with respect to time. Together, they define the direction and speed of motion in the phase plane.
A smaller step size usually gives a more accurate trajectory. A larger step may compute faster, but it can reduce smoothness and numerical accuracy.
Yes. Nonlinear expressions such as x*y, sin(x), or x^2 - y are supported, as long as they use the listed variables and functions.
It is the local trajectory slope dy/dx at the starting point. The calculator computes it from (dy/dt) divided by (dx/dt) when dx/dt is not zero.
Your chosen bounds may be too small, or the system may grow quickly. Increase the x and y limits to inspect more of the solution path.
Grid density controls how many direction arrows appear in the vector field. Higher density shows more detail, while lower density keeps the plot simpler.
The CSV file includes time, x, y, dx/dt, dy/dt, and speed for each computed point. It is useful for analysis, reporting, and reuse.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.