Slope Field & ODE Solver
Read a first-order differential equation as a field of tiny tangent lines. Click the plot to add initial conditions and trace the matching solution curves.
Geometric visualization of differential direction fields 🖖
A slope field represents a first-order ordinary differential equation dy/dx = f(x,y) by drawing short line segments with slope f(x,y) at points on a grid. This provides a geometric visualization of the flow of potential solution curves without solving the equation analytically. It shows equilibrium solutions (where dy/dx = 0), stable/unstable nodes, and inflection points where concavity shifts.
Following the arrows without solving anything 🖖
Think of the slope field as a set of tiny signposts: at every point, one tells you which way a solution must head next. A solution curve simply follows these directions, like a leaf drifting on a current. Because each point has exactly one direction, choosing a starting point (an initial condition) pins down a single curve — click anywhere and the tool traces that unique path for you.
RK4 secretly becomes Simpson's rule 🖖
The RK4 method the tool uses samples the slope four times per step and blends them with weights 1:2:2:1. When your equation is simply dy/dx = f(x), with no y on the right-hand side, that blend collapses exactly into Simpson's rule for integration: h/6·(f₀ + 4f_m + f₁). So numerically solving an ODE and computing a definite integral are, in this case, the very same calculation.