Direction fields, solution curves, and separable equations
A first-order ODE tells you the slope of a solution at every point. Draw those slopes and the solutions almost draw themselves.
Click anywhere on the field to drop an initial condition and watch its solution curve trace along the slopes. Notice that the curves never cross.
Click anywhere on the field to place an initial condition. The solution curve is traced using Euler's method in both forward and backward time.
A first-order ODE relates a function y(x) to its first derivative dy/dx. These equations appear everywhere in science and engineering -- from population dynamics and radioactive decay to electrical circuits and fluid flow. The general form is dy/dx = f(x, y), and the goal is to find a function y(x) that satisfies this relationship.
A direction field (or slope field) draws a small line segment at each point (x, y) with slope f(x, y). These segments show the "flow" of solutions without solving the equation analytically. Click on the field to place initial conditions and watch solution curves emerge, traced by Euler's method.
Key insight: Every point in the plane has a unique slope determined by f(x, y). Solution curves follow these slopes and never cross each other (by the existence and uniqueness theorem). The direction field gives a complete qualitative picture of all possible solutions.
Pick an equation and step through the solution: separate the variables, integrate both sides, and see the family of curves the constant C produces.
Step through the separation of variables process. The solution curves on the right appear when you reach the final step.
A separable equation has the form dy/dx = g(x) · h(y), where the right-hand side factors into a function of x times a function of y. To solve it, we separate variables -- putting all y terms on one side and all x terms on the other -- then integrate both sides independently.
Key insight: Separation of variables converts a differential equation into two ordinary integrals. The integration constant C produces a family of solution curves, each corresponding to a different initial condition. Specifying y(x₀) = y₀ pins down a unique value of C.
Watch dy/dt = ky grow without limit, then toggle the logistic model and see the curve bend into an S as it nears the carrying capacity K.
The exponential model dy/dt = ky has solution y(t) = y₀ e^(kt). Positive k gives exponential growth, negative k gives decay, and k = 0 gives equilibrium.
The simplest first-order ODE, dy/dt = ky, models processes where the rate of change is proportional to the current value. When k > 0 we get exponential growth (populations, compound interest); when k < 0 we get exponential decay (radioactive decay, cooling). Toggle the logistic model to see how a carrying capacity K limits growth in realistic populations.
Key insight: Pure exponential growth is unbounded -- populations would grow forever. The logistic equation dy/dt = ky(1 - y/K) introduces a self-limiting term. As y approaches the carrying capacity K, the growth rate drops to zero, producing the characteristic S-shaped (sigmoid) curve seen in real populations.
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