🌌 Example 81. Quick Slope Check.
For
\begin{equation*}
y' - ty = 2 - t^2\text{,}
\end{equation*}
find the slope at \((t,y)=(-1,6)\text{.}\)
Section 7.1 Slope Fields
When working with first-order differential equations, you can gain insight into solutions without fully solving them. By focusing on how the solutions change, you can use a slope field to visualize the path of any potential solution.
A slope field visually represents the slopes that a solution curve must follow at each point in the plane, shown as short arrows pointing the way a solution would travel—like a leaf carried by a stream.
The pattern created by a slope field provides a visual representation of the family of solutions to the differential equation. A slope field doesn’t show just one solution—it shows them all. From any starting point, a unique curve threads through, always guided by the tiny arrows.
Subsection From Equation to Slope
Every first-order differential equation combines \(t\text{,}\) \(y\text{,}\) and \(y'\text{.}\) By rearranging terms, we can always write it as:
\begin{equation*}
f(t, y, y') = 0,
\end{equation*}
where \(f\) collects all terms. For example:
\begin{equation*}
y' - ty = 2 - t^2 \quad \Rightarrow \quad
\ub{y' - ty - 2 + t^2}_{\large f(t, y, y')} = 0.
\end{equation*}
When it is possible to isolate \(y'\text{,}\) we write:
\begin{equation}
y' = f(t, y).\tag{7.1}
\end{equation}
Here \(f(t,y)\) is the “slope generator”: given any point \((t,y)\text{,}\) it outputs the slope the solution must have there. For instance, if \(t=3\) and \(y=5\text{,}\)
\begin{equation*}
y'(3) = f(3, 5) = \text{some number}.
\end{equation*}
Since \(y'(3)\) is the tangent slope of \(y\) at \(t=3\text{,}\) this number tells you the direction the graph is heading at \((3,5)\text{.}\)
Checkpoint 82. 📖❓ Using the slope generator.
Suppose we have the differential equation \(y' = t + 2y\text{.}\) What is the slope of any solution curve passing through \((t,y) = (1,-2)\text{?}\)
- \(-3\)
- Substitute into \(f(t,y) = t + 2y\text{:}\) \(1 + 2(-2) = -3\text{.}\) That’s the tangent slope there.
- \(3\)
- This would be the slope if y were positive, but here y = -2 changes the result.
- \(-1\)
- Check your math—substitute carefully into \(t + 2y\text{.}\)
- \(1\)
- This would only account for the \(t\) term—don’t forget the \(2y\) part.
Subsection Sketching a Slope Field
To sketch a slope field by hand:
-
Select a small grid of points in the \((t,y)\)-plane.
-
Compute \(f(t,y)\) at each point.
-
Draw a short line segment at the point with that slope.
For example, take
\begin{equation*}
\frac{dy}{dt} = y - t.
\end{equation*}
Computing nine slopes on a \(3\times3\) grid yields:
| \((t , y) \) | \(f(t,y) = y - t \) |
| \((-1 , -1 ) \) | \(-1-(-1)= 0 \) |
| \((-1 , 0 ) \) | \(0-(-1)= 1 \) |
| \((-1 , 1 ) \) | \(1-(-1)= 2 \) |
| \(( 0 , -1 ) \) | \(-1-0= -1 \) |
| \(( 0 , 0 ) \) | \(0-( 0)= 0 \) |
| \(( 0 , 1 ) \) | \(1-0= 1 \) |
| \(( 1 , -1 ) \) | \(-1-1= -2 \) |
| \(( 1 , 0 ) \) | \(0-1= -1 \) |
| \(( 1 , 1 ) \) | \(1-( 1)= 0 \) |
Sketching by hand is great for intuition, but tedious when you need more points. Computer-generated slope fields fill in the gaps, revealing a dense web of arrows that paints the full picture. In Figure 83, the solution curve through \((0,\frac12)\) flows smoothly along the arrows, like an object carried by a current.
A slope field turns an equation into a navigational chart. Each arrow is an instruction every solution must obey. The entire field represents the whole family of solutions, letting you spot patterns in how solutions behave, without ever solving for \(y\) explicitly. Next, we’ll look at autonomous equations, whose slope fields reveal even more structure.
Checkpoint 84. 📖❓ What does a slope field represent?
Which statement best describes what a slope field shows for a first-order differential equation?
- It shows the general flow pattern of any solution.
- A slope field visualizes the slope each solution must take through every point—like a map of directional instructions for all solutions.
- It shows the general flow pattern of a specific solution.
- A slope field doesn’t display one solution—it encodes the entire family of solutions.
- It gives the formulas for all solutions to the equation.
- No formulas appear in a slope field; it’s a picture of slopes, not algebraic expressions.
- It plots the solution curve for any solution.
- A slope field shows the direction a solution must go, not the actual solution curve.
You have attempted of activities on this page.
