Why the Independent Variable Matters
A slope field gives a visual description of a differential equation: at each point, a short line segment shows the slope that a solution curve would have there. In some equations, that slope depends only on the current value of the solution. In others, it also changes with the independent variable. If the independent variable represents time, the second situation describes a rule whose behavior can change as time passes.
For example, the equations \(y'=2-y\) and \(y'=x-y\) have similar forms, but their slope fields behave differently. In the first, every point at the same height \(y\) has the same slope. In the second, the slope at that height can change as \(x\) changes. Recognizing this distinction helps you read the field correctly and avoid treating a time-dependent equation as if its behavior were fixed by \(y\) alone.
Autonomous and Nonautonomous Equations
Consider a first-order differential equation written as \(y'=F(x,y)\), where \(x\) is the independent variable, \(y\) is the dependent variable, and \(F\) assigns a slope to each point in its domain. When the independent variable represents time, it is often written \(t\), giving \(y'=F(t,y)\). The distinction below does not depend on which letter is used.
This is a statement about the function determining the slopes, not merely about how its formula looks. For example, \(F(x,y)=x-x+y\) simplifies to \(y\), so it is autonomous. By contrast, \(F(x,y)=x-y\) genuinely varies with \(x\) when \(y\) is held fixed, so it is nonautonomous. And \(F(x,y)=x\) is also nonautonomous: its slope depends on the independent variable even though it does not depend on \(y\) at all.
A Structural Difference in the Slope Fields
For an autonomous equation \(y'=g(y)\), fix any height \(y=c\). At every point \((x,c)\) in the domain, the slope is \(g(c)\), regardless of \(x\). Thus, slopes are constant as you move horizontally across the field at that height.
For a nonautonomous equation, that horizontal constancy fails at least somewhere: there is a height at which the slope changes as \(x\) changes. The slopes can still be equal at some pairs of points, but they are not determined by height alone. A useful test is therefore to hold \(y\) fixed and evaluate \(F\) at different \(x\)-values.
Justification. If \(F(x,y)=g(y)\), then fixing \(y\) fixes \(g(y)\), so changing \(x\) cannot change the slope. Conversely, suppose \(F(x,y)\) is the same for every \(x\in I\) when \(y\) is fixed. Choose any \(x_0\in I\) and define \(g(y)=F(x_0,y)\). The assumed constancy in \(x\) gives \(F(x,y)=F(x_0,y)=g(y)\) for all \((x,y)\in I\times J\). Hence the equation is autonomous.
Worked Example: Recognizing an Autonomous Field
Worked Example: Recognizing an Autonomous Field
Consider \(y'=2-y\). Determine whether the equation is autonomous, then compare the slopes at several points with the same \(y\)-coordinate.
The slope function is \(F(x,y)=2-y\). It contains no effective dependence on \(x\), and it can be written \(g(y)=2-y\). The equation is therefore autonomous.
At height \(y=1\), the slope at any horizontal position is \(2-1=1\). In particular, at \((0,1)\) it is \(1\), and at \((3,1)\) it is also \(1\).
At height \(y=2\), the slope at any horizontal position is \(2-2=0\). At both \((0,2)\) and \((3,2)\), the slope is \(0\).
Check: Substituting the same \(y\)-value at each pair of points gives the same value of \(2-y\), as the autonomous definition requires.
Worked Example: A Slope That Changes With Both Variables
Worked Example: A Slope That Changes With Both Variables
Consider \(y'=x-y\). Decide whether it is autonomous and calculate slopes at selected points.
Here \(F(x,y)=x-y\). Hold \(y=0\) fixed. At \((x,y)=(-1,0)\), the slope is \(-1-0=-1\). At \((0,0)\), it is \(0-0=0\). At \((1,0)\), it is \(1-0=1\). Since these slopes differ while \(y\) stays fixed, no function of \(y\) alone can describe them. The equation is nonautonomous.
Other rows confirm how the slopes vary. At height \(y=1\), the slopes at \(x=-1,0,1\) are, respectively, \(-1-1=-2\), \(0-1=-1\), and \(1-1=0\). At height \(y=-1\), they are \(-1-(-1)=0\), \(0-(-1)=1\), and \(1-(-1)=2\).
The zero-slope points satisfy \(x-y=0\), or \(y=x\). For instance, the slope is zero at \((0,0)\) and at \((1,1)\). But this is not a horizontal row of zero slopes: the location with zero slope changes as \(x\) changes.
Check: Each listed slope is obtained by substituting its point into \(x-y\). The different slopes along \(y=0\) directly verify nonautonomy.
Worked Example: Dependence on the Independent Variable Alone
Worked Example: Dependence on the Independent Variable Alone
Consider \(y'=x\). Determine its classification and describe the slope at selected points, even though the formula does not contain \(y\).
The slope function is \(F(x,y)=x\). At \(x=-2\), the slope is \(-2\) for every \(y\). At \(x=0\), the slope is \(0\) for every \(y\). At \(x=3\), the slope is \(3\) for every \(y\). In particular, the slopes at \((-2,0)\), \((0,0)\), and \((3,0)\) are \(-2,0,3\), respectively.
Because the slope changes when \(x\) changes, \(F\) cannot be represented as a function of \(y\) alone. Thus \(y'=x\) is nonautonomous, despite having no \(y\) in its right-hand side.
As a check on how these slopes relate to a solution curve, let \(y(x)=\frac{x^2}{2}+C\), where \(C\) is any constant. Differentiation gives \(y'(x)=x\), which is exactly the right-hand side of the equation. Substitution therefore verifies this curve as a solution on any interval where it is defined.
Zero Slopes and Constant Solutions
A point has zero slope when \(F(x,y)=0\). The set of all such points describes where the field segments are horizontal. In an autonomous equation, this condition is \(g(y)=0\), so any root \(y=c\) gives a whole horizontal row of zero slopes. It also gives a constant solution: if \(y(x)=c\), then \(y'=0=g(c)\) at every \(x\).
For a nonautonomous equation, a zero slope at one point does not by itself produce a constant solution. A constant function \(y(x)=c\) is a solution on an interval precisely when \(F(x,c)=0\) for every \(x\) in that interval. The condition must hold throughout the interval, not just at one point.
Worked Example: Checking a Constant Solution in a Nonautonomous Equation
Consider \(y'=x(1-y)\). The equation is nonautonomous because, for example, at \(y=0\) its slope is \(x\), which changes with \(x\). Determine whether \(y=1\) is a constant solution.
For the proposed function \(y(x)=1\), differentiation gives \(y'(x)=0\). Substituting \(y=1\) into the right-hand side gives \(x(1-1)=x\cdot0=0\) for every \(x\). The two sides agree everywhere, so \(y=1\) is a constant solution.
By contrast, \(y=0\) is not a constant solution: its derivative is \(0\), while the right-hand side after substitution is \(x(1-0)=x\), which is not zero throughout an interval. It has zero slope at \(x=0\), but not at other \(x\)-values.
Check: For \(y=1\), both sides equal zero for every \(x\); for \(y=0\), they disagree whenever \(x\ne0\). This verifies the distinction between a zero-slope point and a constant solution.
A Common Classification Mistake
A tempting but incorrect shortcut is to call an equation autonomous whenever its right-hand side does not contain \(y\). That confuses “does not depend on the dependent variable” with “depends only on the dependent variable.” The definition asks whether the slope can be expressed using \(y\) alone.
Another useful caution concerns \(y'=x-y\). Its zero-slope condition \(y=x\) identifies points where the field segments are horizontal. It does not say that \(y=x\) is a solution: the derivative of the curve \(y=x\) is \(1\), whereas the right-hand side \(x-y\) along that curve is \(0\). A proposed solution must satisfy the differential equation, not merely lie on a set of zero-slope points.
Why Time Dependence Matters
When the independent variable represents time, a nonautonomous equation models a rule whose slope can change even when the solution has the same value. For instance, in \(y'=x-y\), the state \(y=0\) has slope \(-1\) at \(x=-1\), slope \(0\) at \(x=0\), and slope \(1\) at \(x=1\). The current value of \(y\) alone does not determine whether the solution is increasing or decreasing; the time coordinate matters too.
This changes how a slope field should be read. In an autonomous field, scanning horizontally across a fixed height always reveals the same slope. In a nonautonomous field, compare actual locations: the slope may change from one column to another, and a zero-slope locus may move across the plane. That observation describes the local direction prescribed by the equation; it does not, by itself, establish how far a solution continues or whether two solution curves can meet.
Check Your Understanding
For each question, distinguish dependence on \(x\) from dependence on \(y\), and justify your answer using the slope function.
- Is \(y'=4y\) autonomous or nonautonomous? What slope occurs at height \(y=2\), regardless of \(x\)?
- Is \(y'=x^2\) autonomous? Compare its slopes at \((0,1)\) and \((2,1)\).
- For \(y'=x-y\), what is the slope at \((2,3)\), and what does that value say about the direction of a solution there?
- For \(y'=x(2-y)\), is \(y=2\) a constant solution? Verify by substitution.
- Why does a zero slope at a single point not establish that a constant function is a solution?
