Home / Learn / Maths / Slope fieldTool #L309 (formerly #1474)

Slope field

See dy/dx = f(x, y) as a field of little slopes and tap anywhere to draw the solution through that point. Growth and decay, Newton's cooling and a mixing tank as stories, plus linear and separable examples with the hand working (separating variables or the integrating factor) and the particular solution for your point.

Keep it one tap away for everyday use.

Bookmark Slope field?

Press Ctrl + D now. Your browser opens its own bookmark box, where you choose the name and folder.

  1. Pick dy/dx
  2. See slopes
  3. Tap a point
  4. Solve it
Nothing is uploadedWorks offline once open

Pick an equation

The slope field

Each little dash has the slope dy/dx at that point. Tap anywhere to draw the solution through it: it follows the dashes. Blue is the newest curve.

Solve it by hand

    Try this

    • Growth: set k below 0 and the same equation describes decay. Every curve has the same shape, just scaled.
    • Cooling: start one curve above room temperature and one below. Both head for Tₛ, the flat line where dT/dt = 0.
    • Mixing tank: whatever salt you start with, the curves all approach c × V grams.
    • dy/dx = −x/y: the curves are circles, and each one stops on the x-axis, where the slope would be infinite.
    • Type your own, such as x - y, sin(x) - y or y*(1 - y), and set the window.

    Check yourself

    Fresh numbers every time. Type a number or a fraction such as 3/4; nothing is saved or sent.

      What it shows

      A differential equation such as dy/dx = x + y does not tell you y; it tells you the slope of the graph of y at every point. Draw a short dash with that slope at many points and you get a slope field (direction field). Every solution is a curve that follows the dashes, and a starting point picks one particular solution out of the whole family.

      This page draws the field, lets you tap to trace solutions through any point, and for the ready-made equations shows how to solve them by hand, either by separating the variables or with an integrating factor, with the constant C worked out for the point you tapped. A dashed green curve shows the formula’s answer on top of the numerical one, so you can see that they agree.

      It follows differential equations in Class 12 maths (order, general and particular solutions, variables separable, linear equations). Slopes and gradients come from the derivative slope tool and areas under curves from riemann sums; for systems of equations, phase portraits and more methods, go on to the ODE solver.

      How to use it

      1. Pick an equation. The stories (growth and decay, Newton’s cooling, a mixing tank) have sliders for their numbers; the linear and separable examples are pure maths. My own… lets you type dy/dx as a formula in x and y, such as x - y or y*(1 - y), with your own window.
      2. Look at the dashes. Where they are flat, dy/dx = 0; where they are steep, y changes fast.
      3. Tap the picture to draw the solution through that point. Each tap adds a curve (the newest is blue). Clear curves starts again.
      4. Under the picture: the start point, the slope there, y further along by Runge-Kutta, and the formula’s value for comparison.
      5. Solve it by hand shows the working for the newest point and the particular solution.

      Worked example

      Pick growth and decay with k = 0.3: dy/dt = 0.3 y, starting at (0, 2). Separating the variables gives dy/y = 0.3 dt, so ln|y| = 0.3t + c and y = C e^(0.3 t). At t = 0, y = 2, so C = 2 and the answer is y = 2 e^(0.3 t). The page shows y at t = 3 as 4.9192 from Runge-Kutta, and the formula gives the same 4.9192.

      Pick the cooling story: tea at 90 °C in a 25 °C room with k = 0.08 per minute follows T = 25 + 65 e^(−0.08 t), which is about 38.1 °C after 20 minutes. In the mixing tank (2 litres a minute of water with 1 g of salt per litre into 100 litres), the salt follows y = 100 − 100 e^(−0.02 t): about 69.9 g after an hour, heading for 100 g.

      Separable or linear?

      • Separable: dy/dx = g(x) h(y). Divide by h(y), multiply by dx, integrate both sides.
      • Linear: dy/dx + P(x) y = Q(x). Multiply by the integrating factor e^(∫P dx); the left side becomes the derivative of y × (integrating factor), so you can integrate.
      • dy/dt = k y and the cooling law are both at once; dy/dx = x + y is linear but not separable.

      Check yourself

      Five fresh questions on growth, cooling, reading a slope from f(x, y), integrating factors and spotting a separable equation. Nothing is saved.

      FAQ

      Why does a curve stop? Where the slope becomes undefined or the curve turns vertical (for dy/dx = −x/y, on the x-axis), a solution y(x) cannot continue, so the drawing stops there.

      How accurate are the curves? They use the fourth-order Runge-Kutta method with small steps; for the ready-made equations the page compares with the exact formula, and the difference is usually under 0.0001.

      Is this from my textbook? It covers the standard school topic with its own examples. It is an independent study aid and is not part of NCERT.

      Where the numbers come from

      • Standard school mathematics: first-order differential equations, slope (direction) fields, separation of variables and linear equations solved with an integrating factor; Newton's law of cooling and a well-stirred mixing tank as models. Written for fibub; no textbook text, figures or exercises are used, and the story numbers are made up.
      • Checked against scipy solve_ivp (rtol 1e-12) for every ready-made equation, and the general solutions of the two linear equations against sympy 1.14 dsolve. The numerical curves use the classical fourth-order Runge-Kutta method.

      Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.

      Last updated 2026-11-09.

      Change log for Slope field

      1. Add #1474 Slope field and solutions Details on GitHub

      Every change to fibub

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