From a Differential Equation to a Curve’s Direction
A first-order differential equation such as \(y'=F(x,y)\) assigns a slope to each point \((x,y)\) in its domain. A slope field displays those instructions as short line segments: the segment at a point has slope \(F(x,y)\).
Why should a curve that solves the differential equation follow those directions? The key is the definition of the derivative. At any point on a differentiable solution curve, the derivative is the limiting slope of nearby secant lines. The equation says exactly what that limiting slope must be.
What It Means to Be a Solution
This definition has two important parts. First, at each input \(x\), the curve has a point \((x,y(x))\). Second, its derivative at that input equals the value of \(F\) at that very point. The field’s direction depends on both coordinates, not just on the curve’s height or just on the input.
A solution is a function, so its graph is a curve with a well-defined tangent slope at each point in its interval. The differential equation specifies that slope. A short segment in a slope field is not a piece of the solution curve; it indicates the direction the curve’s tangent must have if the curve passes through that point.
Why the Derivative Forces the Match
Fix any input \(a\) in the interval of a solution \(y\). The corresponding point on the graph is \((a,y(a))\). For a nonzero increment \(h\) small enough that \(a+h\) remains in the interval, the slope of the secant line between the graph points at \(a\) and \(a+h\) is
By the definition of the derivative, as \(h\) approaches zero these secant slopes approach \(y'(a)\). Since \(y\) is a solution, the differential equation gives \(y'(a)=F(a,y(a))\). Thus the slope specified by the equation at the graph point is precisely the limit of the slopes of the nearby secants.
This conclusion follows directly from the definition of solution and the definition of derivative. It does not say the solution curve must be a straight line. The field gives the tangent direction separately at each point, and that direction can change as the curve moves.
Worked Example: A Curve Following the Field for \(y'=x+y\)
Worked Example: A Curve Following the Field for \(y'=x+y\)
Consider \(y=2e^x-x-1\). Determine the slope prescribed by the equation at the point on this curve where \(x=0\), and check that it matches the curve’s derivative.
First find the point on the curve. At \(x=0\),
The point is \((0,1)\). The differential equation prescribes the slope
Now differentiate the curve:
At \(x=0\), this gives
The values match. More generally, the derivative and the field expression agree along the entire curve:
Substitution verifies the solution, and the point calculation confirms that the segment at \((0,1)\) has the same slope as the curve’s tangent there.
Worked Example: A Positive Slope for \(y'=y\)
Worked Example: A Positive Slope for \(y'=y\)
Consider \(y=3e^x\), and examine its point at \(x=\ln 2\). Find the point and compare the equation’s prescribed slope with the derivative.
Since \(e^{\ln 2}=2\), the curve’s value is
The point is \((\ln 2,6)\). Because \(F(x,y)=y\), the field assigns this point slope
Differentiate the curve and evaluate at the same input:
The slopes agree. To check the curve against the differential equation at every input, substitute its value into the right-hand side:
Thus the curve’s tangent has the field’s positive slope at the selected point, as well as at every other point on the curve.
Worked Example: A Horizontal Tangent for \(y'=x^2-y\)
Worked Example: A Horizontal Tangent for \(y'=x^2-y\)
Consider \(y=x^2-2x+2\), and find the slope at the point on the curve where \(x=1\).
First calculate the height:
The point is \((1,1)\). At this point the differential equation prescribes
Differentiate the curve:
Then evaluate at \(x=1\):
The field segment and curve tangent are both horizontal there. The agreement also holds for every \(x\):
In particular, a zero value of \(F\) means zero derivative at the point, not zero height. Here the curve’s height is \(1\), while its slope is \(0\).
A Local Approximation Explains the Direction
The derivative also describes how the solution’s height changes over a small horizontal step. At a point \(a\), the definition of derivative says that the change in \(y\), divided by the change \(h\) in \(x\), approaches \(y'(a)\) as \(h\) becomes small. For a solution, that limiting rate is \(F(a,y(a))\).
So, for small \(h\), the change in height is approximately the prescribed slope multiplied by the horizontal change:
This is a local description, not an exact formula for every finite step. The slope may change as the point changes, so using one slope for a large movement generally will not describe the whole curve. The field’s short segments communicate local tangent information.
At input \(a\), the solution’s point is \((a,y(a))\).
The differential equation assigns the value \(F(a,y(a))\) as the slope.
The derivative definition makes the curve’s tangent slope \(y'(a)\), and the solution condition gives \(y'(a)=F(a,y(a))\).
Common Pitfall: Confusing Height with Slope
At the point \((1,1)\) in the third example, a tempting but incorrect statement is, “The slope is \(1\), because \(y=1\).” That uses the point’s height as if it were its derivative. But the equation is \(y'=x^2-y\), so both coordinates must be used to find the slope:
The correction is to keep the quantities distinct: \(y(1)\) is the vertical coordinate, while \(y'(1)\) is the tangent slope. Here the point is one unit above the horizontal axis, and its tangent is horizontal.
Another useful distinction is between the field segment and the curve itself. A field segment gives a tangent direction at one point; it does not say the curve follows that same straight direction everywhere. As the curve moves to a new point, the differential equation may assign a different slope.
Why This Connection Matters
A differential equation expresses a rule for change rather than directly stating a formula for the function. The slope field makes that rule visible across the plane. The derivative definition explains why a solution graph must be locally aligned with the field: the curve’s tangent is the limit of its secant directions, and the differential equation prescribes that limiting slope at each point.
This connection is useful when interpreting a differential equation, reading a slope field, or deciding what information a field segment conveys. It also gives a precise way to describe what it means for a curve to follow a field, without claiming that any one segment determines the entire curve.
Check Your Understanding
Answer each question using the connection between the derivative and the field direction.
- For a solution of \(y'=F(x,y)\), what slope does the field assign at the graph point \((a,y(a))\)?
- In the derivative definition, what do the secant slopes between nearby points on a curve approach?
- If a solution passes through \((2,3)\) and satisfies \(y'=x-y\), what slope does the equation prescribe there?
- At a point where \(F(x,y)=0\), what does the equation say about the tangent slope? Does it require the point’s height to be zero?
- Why does one short segment in a slope field not imply that a solution is a straight line?
