π Example 90. Quick Slope Check.
For
\begin{equation*}
y' - ty = 2 - t^2\text{,}
\end{equation*}
find the slope at \((t,y)=(-1,6)\text{.}\)
Section 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 travelβlike 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 doesnβt show just one solutionβit 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{22}
\end{equation}
Here \(f(t,y)\) is the βslope generatorβ: give it any point \((t,y)\) and 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 91. πβ 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 \(f(t,y) = t + 2y\text{:}\) \(1 + 2(-2) = -3\text{.}\) Thatβs the tangent slope there.
- \(3\)
- This would be the slope if y were positive, but here y = -2 changes the result.
- \(-1\)
- Check your mathβsubstitute carefully into \(t + 2y\text{.}\)
- \(1\)
- This would only account for the \(t\) termβdonβt 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:
| \((t , y) \) | \(f(t,y) = y - t \) |
| \((-1 , -1 ) \) | \(-1-(-1)= 0 \) |
| \((-1 , 0 ) \) | \(-1-(-0)= 1 \) |
| \((-1 , 1 ) \) | \(-1-( 1)= 2 \) |
| \(( 0 , -1 ) \) | \(0-(-1)= 1 \) |
| \(( 0 , 0 ) \) | \(0-( 0)= 0 \) |
| \(( 0 , 1 ) \) | \(0-( 1)=-1 \) |
| \(( 1 , -1 ) \) | \(1-(-1)=-2 \) |
| \(( 1 , 0 ) \) | \(1-( 0)=-1 \) |
| \(( 1 , 1 ) \) | \(1-( 1)= 0 \) |
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 FigureΒ 92, 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, weβll look at autonomous equations, whose slope fields reveal even more structure.
π€ Wrap-Up.
ποΈ Key Takeaways...
-
A slope field shows the flow that any solution curve must follow.
-
The function \(f(t,y)\) acts as a slope generator for \(y(t)\) at any point \((t,y)\text{.}\)
-
To sketch a slope field: pick points, compute slopes, and draw short segments.
-
Computer-generated slope fields provide a fuller, denser map and make solution behavior clear even without a formula.
Check Your Understanding.
Checkpoint 93. πβ 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 pointβlike a map of directional instructions for all solutions.
- It shows the general flow pattern of a specific solution.
- A slope field doesnβt display one solutionβit encodes the entire family of solutions.
- It gives the formulas for all solutions to the equation.
- No formulas appear in a slope field; itβs 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.
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