Draw slope fields for first-order ODEs dy/dx=f(x,y). Click to trace solution curves and compare with exact solutions.
A differential equation such as dy/dx = -y can be read before it is solved. At every point of a grid the tool draws a short segment whose slope equals f(x, y): the slope field. Solution curves run tangent to these segments, and clicking anywhere steps a numerical solution curve through the point you chose. Most users are surprised by two things: the traced curves are approximations, not exact solutions, and where f(x, y) is undefined - a division by zero, say - the field goes blank and curves can blow up or stop in ways the picture cannot show.
A concrete input and expected output from the current implementation.
Input
dy/dx = x, then click the point (2, 3)
Expected output
Every segment in the column x = 2 has slope 2, and every segment in the column x = 1 has slope 1; along the y-axis the segments are horizontal. Clicking at (2, 3) traces the parabola y = x^2/2 + 1, which passes through (2, 3) since 4/2 + 1 = 3. Entering y = x^2/2 + 1 as an exact solution overlays the traced curve exactly.
For dy/dx = x the slope depends only on x, so the field repeats down each vertical column and the isoclines are the vertical lines x = k. Antidifferentiating gives the family y = x^2/2 + C; the curve through (2, 3) fixes C = 3 - 2 = 1.
The tool evaluates f(x, y) at each grid point and draws a short segment with that slope, centered on the point. Only the angle varies; length and density are fixed, so a field of steep segments means the derivative is large there. The curves where f(x, y) = k are isoclines - the field's constant-slope lines - and they help you sketch solutions by hand.
Clicking anywhere seeds a solution curve: the tool steps a numerical approximation through the field, using the local segment slope as the derivative at each step, and joins the steps into a curve. Where f is smooth, solution curves of dy/dx = f(x, y) cannot cross, so tracing several starting points shows a family of non-intersecting curves that fill the plane.
Probably neither: the trace is a numerical approximation built from finite steps, so it drifts from the true solution as it moves away from the click point, most where the field changes fast. First make sure the exact curve passes through the point you clicked - a solution is fixed by its initial condition, and curves with different constants never coincide. If they still diverge, the gap is approximation error, not a bug.
The tool draws no segment where f(x, y) is undefined, and solutions often cannot be continued smoothly across such a line. For dy/dx = 1/x the field is empty at x = 0 and the solutions y = ln|x| + C go to negative infinity as x approaches 0 from the right: the gap is the singular point itself. Curves crowd the gap without crossing it.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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