Understanding Direction Fields

Differential Equations · Axiom Academy

LESSON Understanding Direction Fields See the slope of every solution at once — before solving a single equation Take the equation . At any point (x, y) in the plane, that right-hand side is a number — the exact slope a solution curve must have if it ever passes through there. Plug in a point, and f(x, y) hands you the slope on the spot. 2. Constructing a Direction Field Building the field for is completely mechanical — repeat the same three moves at every grid point, and the field assembles itself. Choose (x, y) from a grid spread across the plane. Compute f(x, y) = x - y at that exact point. Sketch a short tilted line with that slope, centered on the point. Do it at every grid point — the field emerges from the pattern. Set x - y = 0 , i.e. y = x : every point on that diagonal has slope 0 — a horizontal segment. Above the line ( y > x ) the slope is negative; below it ( y < x ) the slope is positive. The whole field's tilt flips across that diagonal. Here's the payoff. Solve with y(0) = 2 and you get the exact curve y(x) = x - 1 + 3e^ -x . Follow that curve through the field we just built, and watch what happens at every single point it crosses. 4. Visualizing All Solutions at Once One field, infinitely many solutions. Every initial condition y(0) = y_0 picks out its own curve, and every one of those curves must stay tangent to the very same field. Four starting points give four genuine solutions to below.

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