A Differential Equation Is a Map of Slopes

First, look at the map a differential equation draws. Let's use y'=x−y. This equation sets, at each point (x, y) of the plane, the slope a curve must have there, namely x−y. At the point (2, 0) the slope is 2; at (0, 2) it's −2. So if you draw a short line in that slope direction at every point, the screen fills with little dashes. This is the direction field. Drag the probe around. As the point moves, you see how the slope at that spot is determined, as a number. We haven't found a solution yet. We're just seeing that "which way to go from any point" is already fixed everywhere. A differential equation doesn't hand you the answer directly — it hands you directions like this.

Now follow that map. Drop a start point, take one step in the direction the arrow points, read the new slope there and take another step — follow faithfully like this and a curve gets traced out. That is the solution of the differential equation. Drag the start point around. The curve always runs parallel to the dashes, flowing along the field like a river following the terrain. Whatever point it passes through, the curve's slope there is exactly x−y. You can see that a solution isn't a mystical formula but "the trail left by flowing in the prescribed direction at every moment." This is exactly how a computer solves a differential equation, too: following the arrows in small steps.

One direction field doesn't have one solution curve — it has infinitely many. Look at the screen: the curves fill the whole field like combed hair. Same differential equation, but a different start point follows a different curve. So which of all those curves is "my answer"? The initial condition decides. Drag the handle on the y-axis to change the value of y(0). One condition — "when x=0, y is this much" — picks exactly one out of the dense family of curves. The differential equation gives you the whole family of curves; the initial condition selects one of them. That's why a real problem is "equation + initial condition" as a set: you need the starting point for where it flows to be pinned down to a single answer.

Look separately at one of the most important direction fields: y'=ky, where the slope is proportional to the value of y itself. The key is "the more there is, the faster it changes." When y is large the slope is large; when y is small the slope is small. Drag k and watch. With k positive you get a curve that steepens upward and grows explosively — exponential growth. With k negative it flattens out and dies toward 0 — exponential decay. The solution is y=e^kx. That e from lesson 2 pops out naturally here, because the answer to "something that grows in proportion to its own value" is exactly the exponential function. Population growth, radioactive decay, charging and discharging an RC circuit, compound interest — all of it is the single line y'=ky. This simple direction field is why the exponential shows up so often in nature.

Finally, one way to solve exactly by hand: separation of variables. Look at the direction field of y'=−xy and the arrows circle around the origin; the solutions are concentric circles. How do you confirm this with algebra? Write y' as dy/dx and split the variables to opposite sides, x with x and y with y: y dy = −x dx. Once the variables are separated like this, you can just integrate both sides. y²2 = −x²2 + C, which tidies up to x²+y²=C — circles, exactly. Press the button to unfold this process. The "flow circling around" that the direction field showed is confirmed by separation of variables as the exact formula x²+y²=C. The picture (direction field) and the calculation (separation) point to the same answer. A differential equation whose variables split cleanly is solved with a single integration like this.