Differential Equations: Equations that relate a function $y$ to its derivative $\frac{dy}{dx}$. Instead of giving you $y$ directly, they tell you HOW $y$ changes at every possible coordinate $(x,y)$.
The Slope Field: A grid of tiny line segments. Each segment has exactly the slope specified by the differential equation at that point.
Initial Conditions: There are infinitely many solution curves (due to the $+C$ constant). Tracing a line through a specific $(x_0, y_0)$ point "locks in" the exact mathematical solution!
CLICK ANYWHERE ON THE GRID TO DROP AN INITIAL CONDITION
Select Differential Equation $\frac{dy}{dx}$
Trace Data 0
LATEST INITIAL CONDITION
$x_0 = $No trace
$y_0 = $-
INSTANTANEOUS SLOPE AT $(x,y)$
$\frac{dy}{dx} =$-
Separable DE: The slope $\frac{dy}{dx} = -0.5xy$ depends on BOTH variables. Setting $x=0$ or $y=0$ forces the slope to be zero (horizontal), creating a cross of equilibrium solutions along both axes!