Sketching Slope Fields
Introduced| English |
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| slope field/sləʊp fiːld/ |
A map of slopes across the plane
- A differential equation $\dfrac{dy}{dx}=F(x,y)$ gives a slope at every point.
- Draw a short segment with that slope at each point of a grid, and you get a slope field 斜率场.
- It's a "map" showing which way solution curves must travel everywhere.
- You don't need to solve the equation to draw it — just evaluate the slope.
A slope field for dy/dx = ay
Each dash has slope $\tfrac{dy}{dx}$ at its point — flat near $y=0$ and steeper far away for $\tfrac{dy}{dx}=ay$.
A slope field shows, at each point, the...
Each dash is the tangent direction there.
To draw a slope field you compute slopes only — you do ____ need to solve the equation.
The equation gives the slope directly.
Compute the slope at each point
- At a grid point $(x,y)$, plug into $\dfrac{dy}{dx}=F(x,y)$ to get the number — that's the segment's slope.
- Slope $0$ → a horizontal dash; large positive → a steep upward dash; negative → downward.
- Repeat over a lattice of points to fill the field.
- The equation tells you the slope directly; no antiderivative needed.
For $\dfrac{dy}{dx}=x+y$, find the slope of the dash at the point $(2,3)$.
$2+3=5$.
For $\dfrac{dy}{dx}=xy$, the slope at $(3,2)$ is...
$xy=3\cdot2=6$ (use both coordinates).
Spotting patterns quickly
- If $F$ depends only on $x$: the slope is the same all along a vertical line (columns look identical).
- If $F$ depends only on $y$: the slope is the same all along a horizontal line (rows look identical).
- For $\dfrac{dy}{dx}=y$: slopes grow with height — flat near $y=0$, steep far away.
- Recognizing these patterns speeds up sketching enormously.
If $\dfrac{dy}{dx}$ depends only on $x$, the dashes in each vertical column are parallel.
Same $x$ → same slope down the column.
Equilibrium lines
- Where $F(x,y)=0$, the slope is $0$ — a row (or curve) of horizontal dashes.
- For $\dfrac{dy}{dx}=y$, that happens at $y=0$: a whole horizontal line of flat segments.
- These lines often mark equilibrium behavior — solutions can level off there.
- They're the easiest points to plot, so start with them.
For $\dfrac{dy}{dx}=y$, the dashes are horizontal along the line...
Slope $=y=0$ along $y=0$.
A slope field shows the differential equation's slopes, not the solution curves themselves. The little segments are tangent directions; a solution is a curve that stays tangent to them. And compute the slope with the point's $(x,y)$, using $y$ too when $F$ depends on it — not just $x$.
Sketch a few slopes for $\dfrac{dy}{dx}=x$.
- At $(1,0)$: slope $=1$. At $(2,5)$: slope $=2$. At $(-1,3)$: slope $=-1$.
- Since $F=x$ (no $y$), every point in a vertical column has the same slope.
- Columns of parallel dashes: flat at $x=0$, tilting more as $|x|$ grows.
A slope field draws a short segment of slope $\frac{dy}{dx}=F(x,y)$ at each grid point — a map of tangent directions, no solving needed. Compute the slope at each $(x,y)$; look for patterns (columns match if $F$ depends only on $x$, rows if only on $y$) and horizontal dashes where $F=0$.