integral curve
Every solution of a first-order differential equation can be drawn as a curve in the xy-plane, and there is a vivid way to see what makes it a solution: at every single point along it, the curve runs in exactly the direction the equation prescribes. It never fights the rule; it flows with it. Such a curve — the graph of a solution — is called an integral curve.
Precisely, an integral curve of a first-order equation y' = f(x, y) is the graph of a solution: a curve whose slope at each point (x, y) equals f(x, y), the value the equation assigns there. If you imagine the equation painting a little slope arrow at every point of the plane (the slope field), then an integral curve is a path that stays tangent to those arrows the whole way — like a leaf carried by a current, always pointing downstream. The word 'integral' reflects that you obtain such a curve by, in effect, integrating the slope information.
Integral curves are the geometric soul of a differential equation. The general solution corresponds to a whole family of integral curves filling the plane, and an initial condition picks out the one integral curve passing through a chosen starting point. This viewpoint lets you reason about solutions you cannot write down: even without a formula, you can sketch the integral curves directly from the slope field and see whether solutions rise, fall, level off, or blow up. It is the bridge from algebra to geometry that makes the qualitative theory possible.
For y' = x, the integral curves are the parabolas y = x^2/2 + C. Pick any one, say y = x^2/2: at the point (2, 2) its slope is y' = 2, which is exactly the value f(x, y) = x = 2 the equation demands there. The curve is everywhere tangent to the slope field.
An integral curve hugs the slope field — tangent to the prescribed direction at every point.
Through a point where uniqueness holds, exactly one integral curve passes; integral curves then never cross. Where uniqueness fails they can touch or branch, so non-crossing is a property of well-behaved equations, not a universal law.