1. Calculate: Plug the $x$ and $y$ of each point into the differential equation.
2. Draw: Draw a tiny, straight dash centered at the point with that slope.
3. Scan: Look for rows or columns where the slope is the same. (These are Isoclines!).
4. Surf: Sketch solution curves by following the "Flow" of the dashes.
Complete the slope field for the equation: $ rac{dy}{dx} = x + y$.
Calculate the Dashes: Find the slopes for these points:
$(0,0), (1,0), (0,1), (-1,0), (0,-1), (-1,1), (1,-1)$
Draw the Field: Use the grid below to plot your dashes.
For the equation $ rac{dy}{dx} = y - 2$:
1. At what $y$-value is the slope always Zero?
2. Draw a horizontal line at that value.
3. If you start above that line ($y=3$), do the arrows point up or down?
4. If you start below ($y=1$), where do they point?
If $ rac{dy}{dx} = f(x)$ (no $y$ in the equation)... what do you notice about the dashes in each column? Does the wind direction change as you go up and down, or only as you go left and right?
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On your field from Part I ($dy/dx = x+y$), start at the point $(0, 1)$. Sketch the solution curve that follows the arrows for at least 3 units.
Does it look like a parabola or an exponential curve?
Solve the differential equation $ rac{dy}{dx} = rac{x}{y}$ using **Separation of Variables**.
1. Multiply by $y$ and $dx$ to separate: $y dy = x dx$.
2. Integrate both sides: $\int y dy = \int x dx$.
3. Solve for $y$.
Objective: Explain Slope Fields to a younger student using a pile of leaves and a blower.
The Activity:
1. Point the leaf blower at a pile of leaves.
2. Ask: "If I drop one more leaf here, which way will it go?"
3. "Can we draw an arrow on the ground showing the direction of the blower even if the blower is off?"
The Lesson: "God's Word is the blower. We can draw His 'Direction' on the map of our lives so we always know which way to fly."
Response: ___________________________________________________________