A First Equation: \( y' = y \)

A first-order ODE assigns to each point \( (x,y) \) a slope \( y' = f(x,y) \). A solution is a curve whose tangent matches that slope everywhere.

The simplest interesting case is

\( y' = y, \)

where the slope equals the height. Its solutions are the exponentials

\( y = C e^{x}. \)

One curve passes through each point, filling the plane.

Direction field of \( y'=y \) and the solutions \( y=Ce^{x} \)
Read it geometrically
Below the \( x \)-axis the slopes point down toward it; above, they climb away from it. The field alone already reveals the exponential behaviour.
© Shir Sivroni